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physics
thermodynamics

Materials Science and Engineering An Introduction 8th edition William D. Callister Jr., David G. Rethwisch - Solutions

One integrated circuit design calls for the diffusion of arsenic into silicon wafers; the background concentration of As in Si is 2.5 × 1020 atoms/m3. The predeposition heat treatment is to be conducted at 1000°C for 45 minutes, with a constant surface concentration of 8 × 1026 As atoms/m3. At a
Using mechanics of materials principles (i.e., equations of mechanical equilibrium applied to a free-body diagram), derive Equations 6.4a and 6.4b.
Consider a cylindrical specimen of a steel alloy (Figure 6.21) 10.0 mm (0.39 in.) in diameter and 75 mm (3.0 in.) long that is pulled in tension. Determine its elongation when a load of 20,000 N (4,500 lbf) is applied.
A gray cast iron, the tensile engineering stress-strain curve in the elastic region. Determine the tangent modulus at 10.3 MPa (1500 psi), and (b) the secant modulus taken to 6.9 MPa (1000 psi).
As noted in Section 3.15, for single crystals of some substances, the physical properties are anisotropic; that is, they are dependent on crystallographic direction. One such property is the modulus of elasticity. For cubic single crystals, the modulus of elasticity in a general [uvw] direction,
In Section 2.6 it was noted that the net bonding energy EN between two isolated positive and negative ions is a function of interionic distance r as follows:where A, B, and n are constants for the particular ion pair. Equation 6.25 is also valid for the bonding energy between adjacent ions in solid
Using the solution to Problem 6.13, rank the magnitudes of the moduli of elasticity for the following hypothetical X, Y, and Z materials from the greatest to the least. The appropriate A, B, and n parameters (Equation 6.25) for these three materials are tabulated below; they yield EN in units of
A cylindrical specimen of aluminum having a diameter of 19 mm (0.75 in.) and length of 200 mm (8.0 in.) is deformed elastically in tension with a force of 48,800 N (11,000 lbf). Using the data contained in Table 6.1, determine the following:(a) The amount by which this specimen will elongate in the
A cylindrical bar of steel 10 mm (0.4 in.) in diameter is to be deformed elastically by application of a force along the bar axis. Using the data in Table 6.1, determine the force that will produce an elastic reduction of 3 × 10-3 mm (1.2 × 10-4 in.) in the diameter
A cylindrical specimen of some alloy 8 mm (0.31 in.) in diameter is stressed elastically in tension. A force of 15,700 N (3530 lbf) produces a reduction in specimen diameter of 5 × 10-3 mm (2 × 10-4 in.). Compute Poisson's ratio for this material if its modulus of elasticity is 140 GPa (20.3 ×
A cylindrical specimen of a hypothetical metal alloy is stressed in compression. If its original and final diameters are 20.000 and 20.025 mm, respectively, and its final length is 74.96 mm, compute its original length if the deformation is totally elastic. The elastic and shear moduli for this
Consider a cylindrical specimen of some hypothetical metal alloy that has a diameter of 8.0 mm (0.31 in.). A tensile force of 1000 N (225 lbf) produces an elastic reduction in diameter of 2.8 × 10-4 mm (1.10 × 10-5 in.). Compute the modulus of elasticity for this alloy, given that Poisson's ratio
(a) Equations 6.4a and 6.4b are expressions for normal (σ′) and shear (τ′) stresses, respectively, as a function of the applied tensile stress (σ) and the inclination angle of the plane on which these stresses are taken (θ of Figure 6.4). Make a plot on which is presented the orientation
A brass alloy is known to have a yield strength of 275 MPa (40,000 psi), a tensile strength of 380 MPa (55,000 psi), and an elastic modulus of 103 GPa (15.0 × 106 psi). A cylindrical specimen of this alloy 12.7 mm (0.50 in.) in diameter and 250 mm (10.0 in.) long is stressed in tension and found
A cylindrical metal specimen 12.7 mm (0.5 in.) in diameter and 250 mm (10 in.) long is to be subjected to a tensile stress of 28 MPa (4000 psi); at this stress level the resulting deformation will be totally elastic.(a) If the elongation must be less than 0.080 mm (3.2 × 10-3 in.), which of the
Consider the brass alloy for which the stress-strain behavior is shown. A cylindrical specimen of this material 6 mm (0.24 in.) in diameter and 50 mm (2 in.) long is pulled in tension with a force of 5000 N (1125 lbf). If it is known that this alloy has a Poisson's ratio of 0.30, compute: (a) the
A cylindrical rod 100 mm long and having a diameter of 10.0 mm is to be deformed using a tensile load of 27,500 N. It must not experience either plastic deformation or a diameter reduction of more than 7.5 Ã— 10-3 mm. Of the materials listed as follows, which are possible candidates? Justify
A cylindrical rod 380 mm (15.0 in.) long, having a diameter of 10.0 mm (0.40 in.), is to be subjected to a tensile load. If the rod is to experience neither plastic deformation nor an elongation of more than 0.9 mm (0.035 in.) when the applied load is 24,500 N (5500 lbf), which of the four metals
The tensile engineering stress-strain behavior for a steel alloy. (a) What is the modulus of elasticity? (b) What is the proportional limit? (c) What is the yield strength at a strain offset of 0.002? (d) What is the tensile strength?
A cylindrical specimen of a brass alloy having a length of 60 mm (2.36 in.) must elongate only 10.8 mm (0.425 in.) when a tensile load of 50,000 N (11,240 lbf) is applied. Under these circumstances, what must be the radius of the specimen? Consider this brass alloy to have the stress-strain
load of 85,000 N (19,100 lbf) is applied to a cylindrical specimen of a steel alloy (displaying the stress-strain behavior shown in Figure 6.21) that has a cross-sectional diameter of 15 mm (0.59 in.). (a) Will the specimen experience elastic and/or plastic deformation? Why? (b) If the original
A bar of a steel alloy that exhibits the stress-strain behavior shown in Figure 6.21 is subjected to a tensile load; the specimen is 300 mm (12 in.) long, and of square cross section 4.5 mm (0.175 in.) on a side. (a) Compute the magnitude of the load necessary to produce an elongation of 0.45 mm
A cylindrical specimen of aluminum having a diameter of 0.505 in. (12.8 mm) and a gauge length of 2.000 in. (50.800 mm) is pulled in tension. Use the load-elongation characteristics tabulated below to complete parts (a) through (f).a) Plot the data as engineering stress versus engineering
A specimen of aluminum having a rectangular cross section 10 mm × 12.7 mm (0.4 in. × 0.5 in.) is pulled in tension with 35,500 N (8000 lbf) force, producing only elastic deformation. Calculate the resulting strain.
A specimen of ductile cast iron having a rectangular cross section of dimensions 4.8 mm Ã— 15.9 mm (3/16 in. Ã— 5/8 in.) is deformed in tension. Using the load-elongation data tabulated below, complete problems (a) through (f).a) Plot the data as engineering stress versus
For the titanium alloy, whose stress strain behavior may be observed in the "Tensile Tests" module of Virtual Materials Science and Engineering (VMSE), determine the following: (a) the approximate yield strength (0.002 strain offset), (b) the tensile strength, and (c) the approximate ductility, in
For the tempered steel alloy, whose stress strain behavior may be observed in the "Tensile Tests" module of Virtual Materials Science and Engineering (VMSE), determine the following: (a) the approximate yield strength (0.002 strain offset), (b) the tensile strength, and (c) the approximate
For the aluminum alloy, whose stress strain behavior may be observed in the "Tensile Tests" module of Virtual Materials Science and Engineering (VMSE), determine the following: (a) the approximate yield strength (0.002 strain offset), (b) the tensile strength, and (c) the approximate ductility, in
For the (plain) carbon steel alloy, whose stress strain behavior may be observed in the "Tensile Tests" module of Virtual Materials Science and Engineering (VMSE), determine the following: (a) The approximate yield strength, (b) The tensile strength, and (c) The approximate ductility, in percent
A cylindrical metal specimen having an original diameter of 12.8 mm (0.505 in.) and gauge length of 50.80 mm (2.000 in.) is pulled in tension until fracture occurs. The diameter at the point of fracture is 6.60 mm (0.260 in.), and the fractured gauge length is 72.14 mm (2.840 in.). Calculate the
Calculate the moduli of resilience for the materials having the stress-strain behaviors shown in Figures 6.12 and 6.21.
Determine the modulus of resilience for each of the following alloys:Use modulus of elasticity values in Table 6.1.
A brass alloy to be used for a spring application must have a modulus of resilience of at least 0.75 MPa (110 psi). What must be its minimum yield strength?
Show that Equations 6.18a and 6.18b are valid when there is no volume change during deformation.
A cylindrical specimen of a titanium alloy having an elastic modulus of 107 GPa (15.5 × 106 psi) and an original diameter of 3.8 mm (0.15 in.) will experience only elastic deformation when a tensile load of 2000 N (450 lbf) is applied. Compute the maximum length of the specimen before deformation
Demonstrate that Equation 6.16, the expression defining true strain, may also be represented bywhen specimen volume remains constant during deformation. Which of these two expressions is more valid during necking? Why?
Using the data in Problem 6.28 and Equations 6.15, 6.16, and 6.18a, generate a true stress true strain plot for aluminum. Equation 6.18a becomes invalid past the point at which necking begins; therefore, measured diameters are given below for the last four data points, which should be used in true
A tensile test is performed on a metal specimen, and it is found that a true plastic strain of 0.20 is produced when a true stress of 575 MPa (83,500 psi) is applied; for the same metal, the value of K in Equation 6.19 is 860 MPa (125,000 psi). Calculate the true strain that results from the
For some metal alloy, a true stress of 415 MPa (60,175 psi) produces a plastic true strain of 0.475. How much will a specimen of this material elongate when a true stress of 325 MPa (46,125 psi) is applied if the original length is 300 mm (11.8 in.)? Assume a value of 0.25 for the strain-hardening
The following true stresses produce the corresponding true plastic strains for a brass alloy:What true stress is necessary to produce a true plastic strain of 0.25?
For a brass alloy, the following engineering stresses produce the corresponding plastic engineering strains, prior to necking:On the basis of this information, compute the engineering stress necessary to produce an engineering strain of 0.25.
Find the toughness (or energy to cause fracture) for a metal that experiences both elastic and plastic deformation. Assume Equation 6.5 for elastic deformation, that the modulus of elasticity is 172 GPa (25 × 106 psi), and that elastic deformation terminates at a strain of 0.01. For plastic
For a tensile test, it can be demonstrated that necking begins whenUsing Equation 6.19, determine the value of the true strain at this onset of necking.
Taking the logarithm of both sides of Equation 6.19 yields log σT = log K + n log ∈T Thus, a plot of log σT versus log ∈T in the plastic region to the point of necking should yield a straight line having a slope of n and an intercept (at log σT = 0) of log K. Using the appropriate data
A cylindrical specimen of a brass alloy 7.5 mm (0.30 in.) in diameter and 90.0 mm (3.54 in.) long is pulled in tension with a force of 6000 N (1350 lbf); the force is subsequently released. (a) Compute the final length of the specimen at this time. The tensile stress-strain behavior for this alloy
A steel bar 100 mm (4.0 in.) long and having a square cross section 20 mm (0.8 in.) on an edge is pulled in tension with a load of 89,000 N (20,000 lbf), and experiences an elongation of 0.10 mm (4.0 × 10-3 in.). Assuming that the deformation is entirely elastic, calculate the elastic modulus of
A steel alloy specimen having a rectangular cross section of dimensions 12.7 mm × 6.4 mm (0.5 in. × 0.25 in.) has the stress-strain behavior shown in Figure 6.21. If this specimen is subjected to a tensile force of 38,000 N (8540 lbf) then(a) Determine the elastic and plastic strain values.(b) If
(a) A 10-mm-diameter Brinell hardness indenter produced an indentation 1.62 mm in diameter in a steel alloy when a load of 500 kg was used. Compute the HB of this material. (b) What will be the diameter of an indentation to yield a hardness of 450 HB when a 500 kg load is used?
Estimate the Brinell and Rockwell harnesses for the following: (a) The naval brass for which the stress-strain behavior is shown. (b) The steel alloy for which the stress-strain behavior is shown.
Using the data represented in Figure 6.19, specify equations relating tensile strength and Brinell hardness for brass and nodular cast iron, similar to Equations 6.20a and 6.20b for steels.
Cite five factors that lead to scatter in measured material properties
Below are tabulated a number of Rockwell B hardness values that were measured on a single steel specimen. Compute average and standard deviation hardness values.
Upon what three criteria are factors of safety based?
Determine working stresses for the two alloys that have the stress-strain behaviors..
Consider a cylindrical titanium wire 3.0 mm (0.12 in.) in diameter and 2.5 × 104 mm (1000 in.) long. Calculate its elongation when a load of 500 N (112 lbf) is applied. Assume that the deformation is totally elastic.
For a bronze alloy, the stress at which plastic deformation begins is 275 MPa (40,000 psi), and the modulus of elasticity is 115 GPa (16.7 × 106 psi). (a) What is the maximum load that may be applied to a specimen with a cross-sectional area of 325 mm2 (0.5 in.2) without plastic deformation? (b)
A cylindrical rod of copper (E = 110 GPa, 16 × 106 psi) having a yield strength of 240 MPa (35,000 psi) is to be subjected to a load of 6660 N (1500 lbf). If the length of the rod is 380 mm (15.0 in.), what must be the diameter to allow an elongation of 0.50 mm (0.020 in.)?
Compute the elastic moduli for the following metal alloys, whose stress-strain behaviors may be observed in the "Tensile Tests" module of Virtual Materials Science and Engineering (VMSE): (a) titanium, (b) tempered steel, (c) aluminum, and (d) carbon steel. How do these values compare with those
A large tower is to be supported by a series of steel wires. It is estimated that the load on each wire will be 11,100 N (2500 lbf). Determine the minimum required wire diameter assuming a factor of safety of 2 and a yield strength of 1030 MPa (150,000 psi).
(a) Gaseous hydrogen at a constant pressure of 1.013 MPa (10 atm) is to flow within the inside of a thin-walled cylindrical tube of nickel that has a radius of 0.1 m. The temperature of the tube is to be 300°C and the pressure of hydrogen outside of the tube will be maintained at 0.01013 MPa (0.1
Consider the steady-state diffusion of hydrogen through the walls of a cylindrical nickel tube as described in Problem 6.D2. One design calls for a diffusion flux of 5 × 10-8 mol/m2-s, a tube radius of 0.125 m, and inside and outside pressures of 2.026 MPa (20 atm) and 0.0203 MPa (0.2 atm),
To provide some perspective on the dimensions of atomic defects, consider a metal specimen that has a dislocation density of 104 mm-2. 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
(a) In the manner of Equations 7.1a, 7.1b, and 7.1c, specify the Burgers vector for the simple cubic crystal structure. Its unit cell is shown in Figure 3.24. Also, simple cubic is the crystal structure for the edge dislocation of Figure 4.3, and for its motion as presented. You may also want to
Sometimes cos φ cos λ in Equation 7.2 is termed the Schmid factor. Determine the magnitude of the Schmid factor for an FCC single crystal oriented with its [100] 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 43.1° and 47.9°, respectively, with the tensile axis. If the critical resolved shear stress is 20.7 MPa (3000 psi), will an applied stress of 45 MPa (6500 psi) cause the single
A single crystal of aluminum is oriented for a tensile test such that its slip plane normal makes an angle of 28.1° with the tensile axis. Three possible slip directions make angles of 62.4°, 72.0°, and 81.1° with the same tensile axis. (a) Which of these three slip directions is most
Consider a single crystal of silver 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 1.1 MPa (160 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 [110] direction. If the critical resolved shear stress for this material is 1.75 MPa, calculate the magnitude(s) of applied stress(es) necessary to cause slip to occur on
(a) A single crystal of a metal that has the BCC crystal structure is oriented such that a tensile stress is applied in the [010] direction. If the magnitude of this stress is 2.75 MPa, compute the resolved shear stress in the [111] direction on each of the (110) and (101) planes.(b) On the basis
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 [102] direction. If slip occurs on a (111) plane and in a [101] direction, compute the stress at which the crystal yields if its critical resolved
The critical resolved shear stress for iron is 27 MPa (4000 psi). Determine the maximum possible yield strength for a single crystal of Fe pulled in tension.
List four major differences between deformation by twinning and deformation by slip relative to mechanism, conditions of occurrence, and final result.
Consider two edge dislocations of opposite sign and having slip planes that are separated by several atomic distances as indicated in the diagram. Briefly describe the defect that results when these two dislocations become aligned with each other.
Briefly explain why small-angle grain boundaries are not as effective in interfering with the slip process as are high-angle grain boundaries.
(a) From the plot of yield strength versus (grain diameter)-1/2 for a 70 Cu-30 Zn cartridge brass, determine values for the constants σ0 and ky in Equation 7.7. (b) Now predict the yield strength of this alloy when the average grain diameter is 1.0 × 10-3 mm.
The lower yield point for an iron that has an average grain diameter of 5 × 10-2 mm is 135 MPa (19,500 psi). At a grain diameter of 8 × 10-3 mm, the yield point increases to 260 MPa (37,500 psi). At what grain diameter will the lower yield point be 205 MPa (30,000 psi)?
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, thatif there is no change in specimen volume during the deformation process (i.e., A0l0 = Adld).(b) Using the result of part (a), compute the percent cold work experienced by naval brass (the stress-strain behavior of which is shown in Figure 6.12) when a stress of 400
Two previously under formed 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 16 mm and 11 mm, respectively. The second specimen, with an initial
Two previously under formed 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 as
I it possible for two screw dislocations of opposite sign to annihilate each other? Explain your answer.
A cylindrical specimen of cold-worked copper has a ductility (%EL) of 25%. If its cold worked radius is 10 mm (0.40 in.), what was its radius before deformation?
(a) What is the approximate ductility (%EL) of a brass that has a yield strength of 275 MPa (40,000 psi)? (b) What is the approximate Brinell hardness of a 1040 steel having a yield strength of 690 MPa (100,000 psi)?
Experimentally, it has been observed for single crystals of a number of metals that the critical resolved shear stress Ï„crss is a function of the dislocation density ÏD ashere Ï„0 and A are constants. For copper, the critical resolved shear stress is 2.10 MPa (305
Briefly cite the differences between recovery and recrystallization processes.
Estimate the fraction of recrystallization from the photomicrograph in Figure 7.21c.
(a) What is the driving force for recrystallization? (b) For grain growth?
(a) From Figure 7.25, compute the length of time required for the average grain diameter to increase from 0.01 to 0.1 mm at 500°C for this brass material. (b) Repeat the calculation at 600°C.
The average grain diameter for a brass material was measured as a function of time at 650°C, which is tabulated below at two different times:(a) What was the original grain diameter? (b) What grain diameter would you predict after 150 min at 650°C?
An under formed specimen of some alloy has an average grain diameter of 0.040 mm. You are asked to reduce its average grain diameter to 0.010 mm. Is this possible? If so, explain the procedures you would use and name the processes involved. If it is not possible, explain why.
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.
Grain growth is strongly dependent on temperature (i.e., rate of grain growth increases with increasing temperature), yet temperature is not explicitly given as a part of Equation 7.9. (a) Into which of the parameters in this expression would you expect temperature to be included? (b) On the basis
An uncold-worked brass specimen of average grain size 0.008 mm has a yield strength of 160 MPa (23,500 psi). Estimate the yield strength of this alloy after it has been heated to 600°C for 1000 s, if it is known that the value of ky is 12.0 MPa-mm1/2 (1740 psi-mm1/2).
(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 3.54) for the (100), (110), and (111) planes for FCC.(b) Compare planar densities (Problem 3.55) for the (100), (110), and (111) planes for BCC.
One slip system for the BCC crystal structure is {110}(111). In a manner similar to Figure 7.6b, sketch a {110}-type plane for the BCC structure, representing atom positions with circles. Now, using arrows, indicate two different 111 slip directions within this plane.
One slip system for the HCP crystal structure is {0001}(1120). In a manner similar to Figure 7.6b, sketch a {0001}-type plane for the HCP structure and, using arrows, indicate three different (1120) slip directions within this plane. You might find Figure 3.8 helpful.
Equations 7.1a and 7.1b, expressions for Burgers vectors for FCC and BCC crystal structures, are of the formb = a/2(uvw)where a is the unit cell edge length. Also, since the magnitudes of these Burgers vectors may be determined from the following equation:determine values of |b| for aluminum and
Determine whether or not it is possible to cold work steel so as to give a minimum Brinell hardness of 225, and at the same time have a ductility of at least 12%EL. Justify your decision.
Determine whether or not it is possible to cold work brass so as to give a minimum Brinell hardness of 120 and at the same time have a ductility of at least 20%EL. Justify your decision.

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