MECH 340 Problems For Chapter 6, Fall 2011

6.17 6.18 6.19 6.20 6.21 6.22 Table 6.1, determine the force that will pro- duce an elastic reduction of 4 X 107? mm in the diameter. A cylindrical specimen of an alloy 8 mm in diameter is stressed elastically in tension. A force of 15,700 N produces a reduction in specimen diameter of 5 X 10°*mm. Compute Poisson’s ratio for this material if its modulus of elasticity is 140 GPa. 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 alloy are 105 GPa and 39.7 GPa, respectively. Consider a cylindrical specimen of some hy- pothetical metal alloy that has a diameter of 8.0 mm. A tensile force of 1000 N produces an elastic reduction in diameter of 2.8 X 10™*mm. Compute the modulus of elasticity for this al- loy, given that Poisson’s ratio is 0.30. A brass alloy is known to have a yield strength of 275 MPa, a tensile strength of 380 MPa, and an elastic modulus of 103 GPa. A cylindrical specimen of this alloy 12.7 mm in diameter and 250 mm long is stressed in tension and found to elongate 7.6 mm. On the basis of the infor- mation given, is it possible to compute the mag- nitude of the load that is necessary to produce this change in length? If so, calculate the load. If not, explain why. A cylindrical metal specimen 12.9 mm in di- ameter and 260 mm long is to be subjected to a tensile stress of 28 MPa; at this stress level the resulting deformation will be totally elastic. (a) If the elongation must be less than 0.080 mm, which of the metals in Table 6.1 are suit- able candidates? Why? (b) If,in addition, the maximum permissible di- ameter decrease is 1.2 x 10-7 mm when the ten- sile stress of 28 MPa is applied, which of the metals that satisfy the criterion in part (a) are suitable candidates? Why? Consider the brass alloy for which the stress— strain behavior is shown in Figure 6.12. A’cylin- drical specimen of this material 8 mm in di- Questions and Problems + 191 ameter and 60 mm long is pulled in tension with a force of 5000 N. If it is known that this alloy has a Poisson’s ratio of 0.30, compute (a) the specimen elongation and (b) the reduction in specimen diameter. 6.23 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 experi- ence either plastic deformation or a diameter reduction of more than 7.5 X 107> mm. Of the materials listed as follows, which are possible candidates? Justify your choice(s). Modulus Yield of Elasticity Strength Poisson’s M (GPa) (MPa) __‘ Ratio Aluminum alloy 70 200 0.33 Brass alloy 101 300 0.34 Steel alloy 207 400 0.30 Titanium alloy 107 650 0.34 6.24 A cylindrical rod 380 mm long, having a di- ameter of 10.0 mm, is to be subjected to a ten- sile load. If the rod is to experience neither plastic deformation nor an elongation of more than 0.9 mm when the applied load is 24,500 N, which of the four metals or alloys listed in the following table are possible candidates? Justify your choice(s). Yield Modulus Tensile of Elasticity Strength Strength Material (GPa) (MPa) _(MPa) Aluminum alloy 70 255 420 Brass alloy 100 345 420 Copper 110 250 290 Steel alloy 207 450 550 Tensile Properties 6.25 Figure 6.21 shows 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 off- set of 0.002? (d) What is the tensile strength? 6.26 A cylindrical specimen of a brass alloy having a length of 60 mm must elongate only 10.8 mm 192 - Chapter 6 / Mechanical Properties of Metals 6.27 6.28 6.29 when a tensile load of 50,000 N is applied. Un- der these circumstances, what must be the ra- dius of the specimen? Consider this brass al- loy to have the stress-strain behavior shown in Figure 6.12. A load of 85,000 N 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 20 mm. (a) Will the specimen experience elastic and/or plastic deformation? Why? (b) If the original specimen length is 260 mm, how much will it increase in length when this load is applied? 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 310 mm long and has a square cross section 4.8 mm on a side. (a) Compute the magnitude of the load nec- essary to produce an elongation of 0.48 mm. (b) What will be the deformation after the load has been released? A cylindrical specimen of aluminum having a diameter of 12.8 mm and a gauge length of 50.800 mm is pulled in tension. Use the load-elongation characteristics shown in the following table to complete parts (a) through (f). Load _ Length _N _ mm 0 50.800 7,330 50.851 15,100 50.902 23,100 50.952 30,400 51.003 34,400 51.054 38,400 51.308 41,300 51.816 44,800 52.832 46,200 53.848 47,300 54.864 47,500 55.880 46,100 56.896 44,800 57.658 42,600 58.420 36,400 59.182 Fracture 6.30 6.31 (a) Plot the data as engineering stress versus engineering strain. (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) What is the approximate ductility, in per- cent elongation? (f) Compute the modulus of resilience. A specimen of ductile cast iron having a rec- tangular cross section of dimensions 4.8 mm x 15.9 mm is deformed in tension. Using the load—-elongation data shown in the following table, complete parts (a) through (f). Load Length N mm 0 75.000 4,740 75.025 9,140 75.050 12,920 75.075 16,540 75.113 18,300 75.150 20,170 75.225 22,900 75:375 25,070 75.525 26,800 15:150 28,640 76.500 30,240 78.000 31,100 79.500 31,280 81.000 30,820 82.500 29,180 84.000 27,190 85.500 24,140 87.000 18,970 88.725 Fracture (a) Plot the data as engineering stress versus engineering strain. (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? For the titanium alloy, whose stress-strain be- havior may be observed in the “Tensile Tests” Metal Alloys 6.32 Metal Alloys 6.33 a letal Alloys 6.34 etal Alloys module of Virtual Materials Science and En- gineering (VMSE), determine the following: (a) the approximate yield strength (0.002 strain offset), (b) the tensile strength, and (c) the approximate ductility, in percent elon- gation. How do these values compare with those for the two Ti-6A1-4V alloys presented in Table B.4 of Appendix B? For the tempered steel alloy, whose stress- strain behavior may be observed in the “Ten- sile Tests” module of Virtual Materials Science and Engineering (VMSE), determine the fol- lowing: (a) the approximate yield strength (0.002 strain offset), (b) the tensile strength, and (c) the approximate ductility, in percent elon- gation. How do these values compare with those for the oil-quenched and tempered 4140 and 4340 steel alloys presented in Table B.4 of Appen- dix B? 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 percent elon- gation. How do these values compare with those for the 2024 aluminum alloy (T351 temper) pre- sented in Table B.4 of Appendix B? For the (plain) carbon steel alloy, whose stress- strain behavior may be observed in the “Ten- sile Tests” module of Virtual Materials Science and Engineering (VMSE), determine the fol- lowing: (a) the approximate yield strength, (b) the tensile strength, and (c) the approximate ductility, in percent elon- gation. 6.35 6.36 6.37 6.38 True 6.39 6.40 6.41 Questions and Problems - 193 A cylindrical metal specimen having an orig- inal diameter of 12.9 mm and gauge length of 50.90 mm is pulled in tension until frac- ture occurs. The diameter at the point of frac- ture is 6.70 mm, and the fractured gauge length is 72.14 mm. Calculate the ductility in terms of percent reduction in area and per- cent elongation. Calculate the moduli of resilience for the ma- terials having the stress-strain behaviors shown in Figures 6.12 and 6.21. Determine the modulus of resilience for each of the following alloys: Yield Strength Material MPa Steel alloy 550 Brass alloy 350 Aluminum alloy 250 Titanium alloy 800 Use the modulus of elasticity values in Table 6.1. A brass alloy to be used for a spring applica- tion must have a modulus of resilience of at least 0.80 MPa. What must be its minimum yield strength? Stress and Strain Show that Equations 6.18a and 6.18b are valid when there is no volume change during de- formation. Demonstrate that Equation 6.16, the expres- sion defining true strain, may also be repre- sented by A €r= In (4) when specimen volume remains constant during deformation. Which of these two ex- pressions is more valid during necking? Why? Using the data in Problem 6.29 and Equations 6.15, 6.16, and 6.18a, generate a true stress-true strain plot for aluminum. Equa- tion 6.18a becomes invalid past the point at which necking begins; therefore, measured di- ameters are given in the following table for 194 - 6.42 6.43 6.44 6.45 6.46 the last four data points, which should be used in true stress computations. _ Length _ _ Diameter _ — a ll TM. 56.896 11-71 57.658 10.95 58.420 10.62 59.182 9.40 A tensile test is performed on a metal speci- men, and it is found that a true plastic strain of 0.20 is produced when a true stress of 575 MPa is applied; for the same metal, the value of K in Equation 6.19 is 860 MPa. Calculate the true strain that results from the applica- tion of a true stress of 600 MPa. For some metal alloy, a true stress of 415 MPa produces a plastic true strain of 0.475. How much will a specimen of this material elon- gate when a true stress of 325 MPa is applied if the original length is 300 mm? Assume a value of 0.25 for the strain-hardening expo- nent n. The following true stresses produce the corre- sponding true plastic strains for a brass alloy: True Stress MPa) __ True Strain 345 0.10 15 020 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 en- gineering strains, prior to necking: Engineering Stress ___ (MPa) _ ___ Engineering Strain 235 0.194 250 0.296 On the basis of this information, compute the engineering stress necessary to produce an en- gineering strain of 0.25. Find the toughness (or energy to cause frac- ture) for a metal that experiences both elas- tic and plastic deformation. Assume Equation 6.5 for elastic deformation, that the modulus Chapter 6 / Mechanical Properties of Metals 6.47 6.48 of elasticity is 172 GPa, and that elastic de- formation terminates at a strain of 0.01. For plastic deformation, assume that the relation- ship between stress and strain is described by Equation 6.19, in which the values for K and n are 6900 MPa and 0.30, respectively. Fur- thermore, plastic deformation occurs between strain values of 0.01 and 0.75, at which point fracture occurs. For a tensile test, it can be demonstrated that necking begins when ate or (6.26) der Using Equation 6.19, determine the value of the true strain at this onset of necking. Taking the logarithm of both sides of Equa- tion 6.19 yields logo; = logK + nloger (6.27) Thus, a plot of log a versus log €7 in the plas- tic region to the point of necking should yield a straight line having a slope of and an in- tercept (at log o = 0) of log K. Using the appropriate data tabulated in Problem 6.29, make a plot of log a, versus log ey and determine the values of n and K. Tt will be necessary to convert engineering stresses and strains to true stresses and strains using Equations 6.18a and 6.18b. Elastic Recovery after Plastic Deformation 6.49 6.50 A cylindrical specimen of a brass alloy 7.8 mm in diameter and 95.0 mm long is pulled in ten- sion with a force of 6000 N; the force is subse- quently released. (a) Compute the final length of the specimen at this time. The tensile stress-strain behavior for this alloy is shown in Figure 6.12. (b) Compute the final specimen length when the load is increased to 16,500 N and then re- leased. A steel alloy specimen having a rectangular cross section of dimensions 12.7 mm x 6.4 mm has the stress-strain behavior shown in Figure 6.21. If this specimen is subjected to a tensile force of 38,000 N, then (a) Determine the elastic and plastic strain values.