Concepts of Stress and Strain


 Esmond Jordan
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1 CHAPTER 6 MECHANICAL PROPERTIES OF METALS PROBLEM SOLUTIONS Concepts of Stress and Strain 6.4 A cylindrical specimen of a titanium alloy having an elastic modulus of 107 GPa ( psi) and an original diameter of 3.8 mm (0.15 in.) will experience only elastic deformation when a tensile load of 000 N (450 lb f ) is applied. Compute the maximum length of the specimen before deformation if the maximum allowable elongation is 0.4 mm ( in.). We are asked to compute the maximum length of a cylindrical titanium alloy specimen (before deformation) that is deformed elastically in tension. For a cylindrical specimen A 0 = π d 0 where d 0 is the original diameter. Combining Equations 6.1, 6., and 6.5 and solving for l 0 leads to l 0 = l ε = l Ê= σ E l E Ê= F A 0 d l Eπ 0 F = l Eπ d 0 4F = ( m)( N /m ) (π)( m) (4)(000 N) = 0.55 m = 55 mm (10.0 in.)
2 6.6 Consider a cylindrical titanium wire 3.0 mm (0.1 in.) in diameter and mm (1000 in.) long. Calculate its elongation when a load of 500 N (11 lb f ) is applied. Assume that the deformation is totally elastic. In order to compute the elongation of the Ti wire when the 500 N load is applied we must employ Equations 6.1, 6., and 6.5. Solving for l and realizing that for Ti, E = 107 GPa ( psi) (Table 6.1), l = l 0 ε = l 0 σ E = l 0 F EA 0 = l 0 F Eπ d = 4l 0 F Eπd 0 0 = (4)(5 m)(500 N) ( N/m )(π )( = m = 16.5 mm (0.65 in.) m)
3 6.11 Figure 6. shows, for a gray cast iron, the tensile engineering stress strain curve in the elastic region. Determine (a) the tangent modulus at 10.3 MPa (1500 psi), and (b) the secant modulus taken to 6.9 MPa (1000 psi). (a) This portion of the problem asks that the tangent modulus be determined for the gray cast iron, the stressstrain behavior of which is shown in Figure 6.. In the figure below is shown a tangent draw on the curve at a stress of 10.3 MPa (1500 psi). The slope of this line (i.e., σ/ ε), the tangent modulus, is computed as follows: σ ε = 15 MPa 5 MPa = 1410 MPa = 1.41 GPa ( psi) (b) The secant modulus taken from the origin is calculated by taking the slope of a secant drawn from the origin through the stressstrain curve at 6.9 MPa (1,000 psi). This secant is drawn on the curve shown below:
4 The slope of this line (i.e., σ/ ε), the secant modulus, is computed as follows: σ ε = 15 MPa 0 MPa = 3190 MPa = 3.19 GPa ( psi) 6.4 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 4,500 N (5500 lb f ), which of the four metals or alloys listed below are possible candidates? Justify your choice(s). Material Modulus of Elasticity (GPa) Yield Strength (MPa) Tensile Strength (MPa) Aluminum alloy Brass alloy Copper Steel alloy This problem asks that we ascertain which of four metal alloys will not (1) experience plastic deformation, and () elongate more than 0.9 mm when a tensile load of 4,500 N is applied. It is first necessary to compute the
5 stress using Equation 6.1; a material to be used for this application must necessarily have a yield strength greater than this value. Thus, σ = F A 0 = 4,500 N π m = 31 MPa Of the metal alloys listed, only brass and steel have yield strengths greater than this stress. Next, we must compute the elongation produced in both brass and steel using Equations 6. and 6.5 in order to determine whether or not this elongation is less than 0.9 mm. For brass l = σ l 0 E (31 MPa)(380 mm) = MPa = 1.19 mm Thus, brass is not a candidate. However, for steel l = σ l 0 E (31 MPa)(380 mm) = MPa = 0.57 mm Therefore, of these four alloys, only steel satisfies the stipulated criteria.
6 Tensile Properties 6.9 A cylindrical specimen of aluminum having a diameter of in. (1.8 mm) and a gauge length of.000 in. ( mm) is pulled in tension. Use the load elongation characteristics tabulated below to complete parts (a) through (f). Load Length N lb f mm in ,330 1, ,100 3, ,100 5, ,400 6, ,400 7, ,400 8, ,300 9, ,800 10, ,00 10, ,300 10, ,500 10, ,100 10, ,800 10, ,600 9, ,400 8, 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 (d) Determine the tensile strength of this alloy. (e) What is the approximate ductility, in percent elongation? (f) Compute the modulus of resilience.
7 This problem calls for us to make a stressstrain plot for aluminum, given its tensile loadlength data, and then to determine some of its mechanical characteristics. (a) The data are plotted below on two plots: the first corresponds to the entire stressstrain curve, while for the second, the curve extends to just beyond the elastic region of deformation. (b) The elastic modulus is the slope in the linear elastic region (Equation 6.10) as
8 E = σ ε = 00 MPa 0 MPa = MPa = 6.5 GPa ( psi) (c) For the yield strength, the 0.00 strain offset line is drawn dashed. It intersects the stressstrain curve at approximately 85 MPa (41,000 psi ). (d) The tensile strength is approximately 370 MPa (54,000 psi), corresponding to the maximum stress on the complete stressstrain plot. (e) The ductility, in percent elongation, is just the plastic strain at fracture, multiplied by onehundred. The total fracture strain at fracture is 0.165; subtracting out the elastic strain (which is about 0.005) leaves a plastic strain of Thus, the ductility is about 16%EL. (f) From Equation 6.14, the modulus of resilience is just U r = σ y E which, using data computed above gives a value of U r = (85 MPa) ()( MPa) = 0.65 MN/m = N/m = J/m 3 (93.8 in. lb f /in. 3 )
9 6.35 A cylindrical metal specimen having an original diameter of 1.8 mm (0.505 in.) and gauge length of mm (.000 in.) is pulled in tension until fracture occurs. The diameter at the point of fracture is 6.60 mm (0.60 in.), and the fractured gauge length is 7.14 mm (.840 in.). Calculate the ductility in terms of percent reduction in area and percent elongation. This problem calls for the computation of ductility in both percent reduction in area and percent elongation. Percent reduction in area is computed using Equation 6.1 as %RA = π d 0 π d f π d in which d 0 and d f are, respectively, the original and fracture crosssectional areas. Thus, %RA = 1.8 mm 6.60 mm π π 1.8 mm π 100 = 73.4% While, for percent elongation, we use Equation 6.11 as %EL = l f l l 0 = 7.14 mm mm mm 100 = 4% 6.48 Taking the logarithm of both sides of Equation 6.19 yields log σ T = log K + n log T (6.7) Thus, a plot of log σ T 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 σ T = 0) of log K.
10 Using the appropriate data tabulated in Problem 6.9, make a plot of log σ T versus log T and determine the values of n and K. It will be necessary to convert engineering stresses and strains to true stresses and strains using Equations 6.18a and 6.18b. This problem calls for us to utilize the appropriate data from Problem 6.9 in order to determine the values of n and K for this material. From Equation 6.7 the slope and intercept of a log σ T versus log ε T plot will yield n and log K, respectively. However, Equation 6.19 is only valid in the region of plastic deformation to the point of necking; thus, only the 7th, 8th, 9th, and 10th data points may be utilized. The loglog plot with these data points is given below. The slope yields a value of for n, whereas the intercept gives a value of.7497 for log K, and thus K = = 56 MPa.
11 Elastic Recovery After Plastic Deformation 6.50 A steel alloy specimen having a rectangular cross section of dimensions 1.7 mm 6.4 mm (0.5 in. 0.5 in.) has the stress strain behavior shown in Figure 6.1. If this specimen is subjected to a tensile force of 38,000 N (8540 lb f ) then (a) Determine the elastic and plastic strain values. (b) If its original length is 460 mm (18.0 in.), what will be its final length after the load in part (a) is applied and then released? (a) We are asked to determine both the elastic and plastic strain values when a tensile force of 38,000 N (8540 lb f ) is applied to the steel specimen and then released. First it becomes necessary to determine the applied stress using Equation 6.1; thus σ = F A 0 = F b 0 d 0 where b 0 and d 0 are crosssectional width and depth (1.7 mm and 6.4 mm, respectively). Thus σ = 38,000 N ( m)( m) = N /m = 468 MPa (68,300 psi) From Figure 6.1, this point is in the plastic region so the specimen will be both elastic and plastic strains. The total strain at this point, ε t, is about We are able to estimate the amount of permanent strain recovery ε e from Hooke's law, Equation 6.5 as ε e = σ E And, since E = 07 GPa for steel (Table 6.1) ε e = 468 MPa MPa = The value of the plastic strain, ε p is just the difference between the total and elastic strains; that is
12 ε p = ε t ε e = = (b) If the initial length is 460 mm (18.0 in.) then the final specimen length l i may be determined from a rearranged form of Equation 6. using the plastic strain value as l i = l 0 (1 + ε p ) = (460 mm)( ) = mm (18.14 in.) Hardness 6.53 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 6.0a and 6.0b for steels. These equations, for a straight line, are of the form TS = C + (E)(HB) where TS is the tensile strength, HB is the Brinell hardness, and C and E are constants, which need to be determined. One way to solve for C and E is analyticallyestablishing two equations using TS and HB data points on the plot, as (TS) 1 = C + (E)(BH) 1 (TS) = C + (E)(BH) Solving for E from these two expressions yields E = (TS) 1 (TS) (HB) (HB) 1 For nodular cast iron, if we make the arbitrary choice of (HB) 1 and (HB) as 00 and 300, respectively, then, from Figure 6.19, (TS) 1 and (TS) take on values of 600 MPa (87,000 psi) and 1100 MPa (160,000 psi), respectively. Substituting these values into the above expression and solving for E gives
13 E = 600 MPa 1100 MPa 00 HB 300 HB = 5.0 MPa/HB (730 psi/hb) Now, solving for C yields C = (TS) 1 (E)(BH) 1 = 600 MPa  (5.0 MPa/HB)(00 HB) = 400 MPa ( 59,000 psi) Thus, for nodular cast iron, these two equations take the form TS(MPa) = x HB TS(psi) = 59, x HB Now for brass, we take (HB) 1 and (HB) as 100 and 00, respectively, then, from Figure 7.31, (TS) 1 and (TS) take on values of 370 MPa (54,000 psi) and 660 MPa (95,000 psi), respectively. Substituting these values into the above expression and solving for E gives E = 370 MPa 660 MPa 100 HB 00 HB =.9 MPa/HB (410 psi/hb) Now, solving for C yields C = (TS) 1 (E)(BH) 1 = 370 MPa (.9 MPa/HB)(100 HB) = 80 MPa (13,000 psi) Thus, for brass these two equations take the form TS(MPa) = x HB TS(psi) = 13, x HB
14 Variability of Material Properties 6.55 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 The average of the given hardness values is calculated using Equation 6.1 as HRB = 15 HRB i i=1 15 = = 85.3 And we compute the standard deviation using Equation 6. as follows: s = 15 i=1 ( HRB i HRB) 15 1 = ( ) + ( ) ( ) 1/ 14 = =.08
15 DESIGN PROBLEMS 6.D1 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 (500 lb f ). Determine the minimum required wire diameter assuming a factor of safety of and a yield strength of 1030 MPa (150,000 psi). For this problem the working stress is computed using Equation 6.4 with N =, as σ w = σ y = 1030 MPa = 515 MPa (75, 000 psi ) Since the force is given, the area may be determined from Equation 6.1, and subsequently the original diameter d 0 may be calculated as A 0 = F = π d 0 σ w And d 0 = 4F π σ w = (4)(11,100 N) π( N /m ) = m = 5.3 mm (0.06 in.)
16 6.D (a) Gaseous hydrogen at a constant pressure of MPa (10 atm) is to flow within the inside of a thinwalled 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 MPa (0.1 atm). Calculate the minimum wall thickness if the diffusion flux is to be no greater than mol/m s. The concentration of hydrogen in the nickel, C H (in moles hydrogen per m 3 of Ni) is a function of hydrogen pressure, P H (in MPa) and absolute temperature (T) according to 1.3 kj/mol C H = 30.8 p H exp (6.8) RT Furthermore, the diffusion coefficient for the diffusion of H in Ni depends on temperature as D H = exp kj/mol RT (6.9) (b) For thinwalled cylindrical tubes that are pressurized, the circumferential stress is a function of the pressure difference across the wall (Δp), cylinder radius (r), and tube thickness (Δx) as σ = r p 4 x (6.30) Compute the circumferential stress to which the walls of this pressurized cylinder are exposed. (c) The roomtemperature yield strength of Ni is 100 MPa (15,000 psi) and, furthermore, σ y diminishes about 5 MPa for every 50 C rise in temperature. Would you expect the wall thickness computed in part (b) to be suitable for this Ni cylinder at 300 C? Why or why not? (d) If this thickness is found to be suitable, compute the minimum thickness that could be used without any deformation of the tube walls. How much would the diffusion flux increase with this reduction in thickness? On the other hand, if the thickness determined in part (c) is found to be unsuitable, then specify a minimum thickness that you would use. In this case, how much of a diminishment in diffusion flux would result? (a) This portion of the problem asks for us to compute the wall thickness of a thinwalled cylindrical Ni tube at 300 C through which hydrogen gas diffuses. The inside and outside pressures are, respectively, and MPa, and the diffusion flux is to be no greater than mol/m s. This is a steadystate diffusion problem, which necessitates that we employ Equation 5.3. The concentrations at the inside and outside wall faces may be determined using Equation 6.8, and, furthermore, the diffusion coefficient is computed using Equation 6.9. Solving for x (using Equation 5.3) x = D C J
17 = mol/m s ( ,560 J /mol ) exp (8.31 J/mol  K)( K) 1,300 J/mol (30.8) exp MPa MPa (8.31 J/mol  K)( K) ( ) = m =.5 mm (b) Now we are asked to determine the circumferential stress: σ = r p 4 x = (0.10 m)(1.013 MPa MPa) (4)(0.005 m) = 10.0 MPa (c) Now we are to compare this value of stress to the yield strength of Ni at 300 C, from which it is possible to determine whether or not the.5 mm wall thickness is suitable. From the information given in the problem, we may write an equation for the dependence of yield strength (σ y ) on temperature (T) as follows: σ y = 100 MPa 5 MPa ( 50 C T T r) where T r is room temperature and for temperature in degrees Celsius. Thus, at 300 C σ y = 100 MPa (0.1 MPa/ C) (300 C 0 C) = 7 MPa Inasmuch as the circumferential stress (10 MPa) is much less than the yield strength (7 MPa), this thickness is entirely suitable.
18 (d) And, finally, this part of the problem asks that we specify how much this thickness may be reduced and still retain a safe design. Let us use a working stress by dividing the yield stress by a factor of safety, according to Equation 6.4. On the basis of our experience, let us use a value of.0 for N. Thus σ w = σ y N 7 MPa = = 36 MPa Using this value for σ w and Equation 6.30, we now compute the tube thickness as x = r p 4σ w = (0.10 m)(1.013 MPa MPa) 4(36 MPa) = m = 0.70 mm Substitution of this value into Fick's first law we calculate the diffusion flux as follows: J = D C x = ( ,560 J/mol ) exp (8.31 J/mol  K)( K) 1,300 J /mol (30.8) exp (8.31 J/mol  K)( K) MPa MPa m ( ) = mol/m s Thus, the flux increases by approximately a factor of 3.5, from to mol/m s with this reduction in thickness.
19 14:440:407 Fall 010 Additional problems and SOLUTION OF HOMEWORK 07 DISLOCATIONS AND STRENGTHENING MECHANISMS PROBLEM SOLUTIONS Basic Concepts of Dislocations Characteristics of Dislocations 7.1 To provide some perspective on the dimensions of atomic defects, consider a metal specimen that has a dislocation density of 10 4 mm . Suppose that all the dislocations in 1000 mm 3 (1 cm 3 ) were somehow removed and linked end to end. How far (in miles) would this chain extend? Now suppose that the density is increased to mm  by cold working. What would be the chain length of dislocations in 1000 mm 3 of material? The dislocation density is just the total dislocation length per unit volume of material (in this case per cubic millimeters). Thus, the total length in 1000 mm 3 of material having a density of 10 4 mm  is just (10 4 mm  )(1000 mm 3 ) = 10 7 mm = 10 4 m = 6. mi Similarly, for a dislocation density of mm , the total length is (10 10 mm  )(1000 mm 3 ) = mm = m = mi
20 7. 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. When the two edge dislocations become aligned, a planar region of vacancies will exist between the dislocations as:
21 answer. 7.3 Is it possible for two screw dislocations of opposite sign to annihilate each other? Explain your It is possible for two screw dislocations of opposite sign to annihilate one another if their dislocation lines are parallel. This is demonstrated in the figure below.
22 7.4 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. For the various dislocation types, the relationships between the direction of the applied shear stress and the direction of dislocation line motion are as follows: edge dislocationparallel screw dislocationperpendicular mixed dislocationneither parallel nor perpendicular
23 Slip Systems 7.5 (a) Define a slip system. (b) Do all metals have the same slip system? Why or why not? (a) A slip system is a crystallographic plane, and, within that plane, a direction along which dislocation motion (or slip) occurs. (b) All metals do not have the same slip system. The reason for this is that for most metals, the slip system will consist of the most densely packed crystallographic plane, and within that plane the most closely packed direction. This plane and direction will vary from crystal structure to crystal structure.
24 7.6 (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. (a) For the FCC crystal structure, the planar density for the (110) plane is given in Equation 3.11 as PD 110 (FCC) = 1 4 R = R Furthermore, the planar densities of the (100) and (111) planes are calculated in Homework Problem 3.54, which are as follows: PD 100 (FCC) = 1 4 R = 0.5 R PD 111 (FCC) = 1 R 3 = 0.9 R (b) For the BCC crystal structure, the planar densities of the (100) and (110) planes were determined in Homework Problem 3.55, which are as follows: PD 100 (BCC) = PD 110 (BCC) = 3 16R = 0.19 R 3 8 R = 0.7 R Below is a BCC unit cell, within which is shown a (111) plane.
25 (a) The centers of the three corner atoms, denoted by A, B, and C lie on this plane. Furthermore, the (111) plane does not pass through the center of atom D, which is located at the unit cell center. The atomic packing of this plane is presented in the following figure; the corresponding atom positions from the Figure (a) are also noted. (b) Inasmuch as this plane does not pass through the center of atom D, it is not included in the atom count. One sixth of each of the three atoms labeled A, B, and C is associated with this plane, which gives an equivalence of onehalf atom. In Figure (b) the triangle with A, B, and C at its corners is an equilateral triangle. And, from Figure (b), the area of this triangle is xy. The triangle edge length, x, is equal to the length of a face diagonal, as indicated in Figure (a). And its length is related to the unit cell edge length, a, as
26 x = a + a = a or x = a For BCC, a = 4 R 3 (Equation 3.3), and, therefore, x = 4R 3 Also, from Figure (b), with respect to the length y we may write y + x = x which leads to y = x 3. And, substitution for the above expression for x yields y = x 3 = 4 R 3 3 = 4 R Thus, the area of this triangle is equal to AREA = 1 x y = 1 4 R 4 R 3 = 8 R 3 And, finally, the planar density for this (111) plane is PD 111 (BCC) = 0.5 atom 3 8 R = 16 R = 0.11 R 3
27 7.7 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. Below is shown the atomic packing for a BCC { 110} type plane. The arrows indicate two different type directions.
28 7.8 One slip system for the HCP crystal structure is { 0001} 110. In a manner similar to Figure 7.6b, sketch a { 0001}type plane for the HCP structure and, using arrows, indicate three different 110 slip directions within this plane. You might find Figure 3.8 helpful. Below is shown the atomic packing for an HCP type directions. { 0001} type plane. The arrows indicate three different
29 the form 7.9 Equations 7.1a and 7.1b, expressions for Burgers vectors for FCC and BCC crystal structures, are of b = a uvw where a is the unit cell edge length. Also, since the magnitudes of these Burgers vectors may be determined from the following equation: b = a ( u + v + w ) 1/ (7.10) determine values of b for aluminum and chromium. You may want to consult Table 3.1. For Al, which has an FCC crystal structure, R = nm (Table 3.1) and a = (Equation 3.1); also, from Equation 7.1a, the Burgers vector for FCC metals is R = nm b = a 110 Therefore, the values for u, v, and w in Equation 7.10 are 1, 1, and 0, respectively. Hence, the magnitude of the Burgers vector for Al is b = a u + v + w = nm (1 ) + (1 ) + (0) = 0.86 nm For Cr which has a BCC crystal structure, R = nm (Table 3.1) and a = 4 R 3 = nm (Equation 3.3); also, from Equation 7.1b, the Burgers vector for BCC metals is b = a 111 Therefore, the values for u, v, and w in Equation 7.10 are 1, 1, and 1, respectively. Hence, the magnitude of the Burgers vector for Cr is b = nm (1) + (1) + (1) = nm
30 7.10 (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 3.4. Also, simple cubic is the crystal structure for the edge dislocation of Figure 4.3, 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 7.10, formulate an expression for the magnitude of the Burgers vector, b, for simple cubic. (a) This part of the problem asks that we specify the Burgers vector for the simple cubic crystal structure (and suggests that we consult the answer to Concept Check 7.1). This Concept Check asks that we select the slip system for simple cubic from four possibilities. The correct answer is { 100} 010. Thus, the Burgers vector will lie in a 010 type direction. Also, the unit slip distance is a (i.e., the unit cell edge length, Figures 4.3 and 7.1). Therefore, the Burgers vector for simple cubic is b = a 010 Or, equivalently b = a 100 (b) The magnitude of the Burgers vector, b, for simple cubic is b = a( ) 1/ = a
31 Slip in Single Crystals 7.11 Sometimes cos φ cos λ in Equation 7. 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. We are asked to compute the Schmid factor for an FCC crystal oriented with its [100] direction parallel to the loading axis. With this scheme, slip may occur on the (111) plane and in the [110] direction as noted in the figure below. The angle between the [100] and [11 0] directions, λ, may be determined using Equation 7.6 λ = cos 1 u 1 u + v 1 v + w 1 w u 1 + v1 + w1 ( ) u + v + w ( ) where (for [100]) u 1 = 1, v 1 = 0, w 1 = 0, and (for [11 0]) u = 1, v = 1, w = 0. Therefore, λ is equal to λ = cos 1 (1)(1) + (0)( 1) + (0)(0) (1) + (0) + (0) [ ][(1) + ( 1) + (0) ] = cos 1 1 = 45
32 Now, the angle φ is equal to the angle between the normal to the (111) plane (which is the [111] direction), and the [100] direction. Again from Equation 7.6, and for u 1 = 1, v 1 = 1, w 1 = 1, and u = 1, v = 0, and w = 0, we have φ = cos 1 (1)(1) + (1)(0) + (1)(0) (1) + (1) + (1) [ ][(1) + (0) + (0) ] = cos = 54.7 Therefore, the Schmid factor is equal to cos λ cos φ = cos (45 ) cos (54.7 ) = 1 1 =
33 7.1 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 0.7 MPa (3000 psi), will an applied stress of 45 MPa (6500 psi) cause the single crystal to yield? If not, what stress will be necessary? This problem calls for us to determine whether or not a metal single crystal having a specific orientation and of given critical resolved shear stress will yield. We are given that φ = 43.1, λ = 47.9, and that the values of the critical resolved shear stress and applied tensile stress are 0.7 MPa (3000 psi) and 45 MPa (6500 psi), respectively. From Equation 7. τ R = σ cos φ cos λ = (45 MPa)(cos 43.1 )(cos 47.9 ) =.0 MPa (3181 psi) Since the resolved shear stress ( MPa) is greater than the critical resolved shear stress (0.7 MPa), the single crystal will yield.
34 7.13 A single crystal of aluminum is oriented for a tensile test such that its slip plane normal makes an angle of 8.1 with the tensile axis. Three possible slip directions make angles of 6.4, 7.0, and 81.1 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 1.95 MPa (80 psi), determine the critical resolved shear stress for aluminum. We are asked to compute the critical resolved shear stress for Al. As stipulated in the problem, φ = 8.1, while possible values for λ are 6.4, 7.0, and (a) Slip will occur along that direction for which (cos φ cos λ) is a maximum, or, in this case, for the largest cos λ. Cosines for the possible λ values are given below. cos(6.4 ) = 0.46 cos(7.0 ) = 0.31 cos(81.1 ) = 0.15 Thus, the slip direction is at an angle of 6.4 with the tensile axis. (b) From Equation 7.4, the critical resolved shear stress is just τ crss = σ y (cos φ cos λ) max = (1.95 MPa) [ cos (8.1 ) cos (6.4 )] = 0.80 MPa (114 psi)
35 7.14 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 [1 01] direction, and is initiated at an applied tensile stress of 1.1 MPa (160 psi), compute the critical resolved shear stress. This problem asks that we compute the critical resolved shear stress for silver. In order to do this, we must employ Equation 7.4, but first it is necessary to solve for the angles λ and φ which are shown in the sketch below. The angle λ is the angle between the tensile axis i.e., along the [001] direction and the slip direction i.e., [1 01]. The angle λ may be determined using Equation 7.6 as λ = cos 1 u 1 u + v 1 v + w 1 w u 1 + v1 + w1 ( ) u + v + w ( ) where (for [001]) u 1 = 0, v 1 = 0, w 1 = 1, and (for [1 01]) u = 1, v = 0, w = 1. Therefore, λ is equal to λ = cos 1 (0)( 1) + (0)(0) + (1)(1) (0) + (0) + (1) [ ][( 1) + (0) + (1) ] = cos 1 1 = 45
36 Furthermore, φ is the angle between the tensile axis the [001] direction and the normal to the slip plane i.e., the (111) plane; for this case this normal is along a [111] direction. Therefore, again using Equation 7.6 φ = cos 1 (0)(1) + (0)(1) + (1)(1) (0) + (0) + (1) [ ][(1) + (1) + (1) ] = cos = 54.7 And, finally, using Equation 7.4, the critical resolved shear stress is equal to τ crss = σ y (cos φ cos λ) [ ] = (1.1 MPa) = (1.1 MPa) cos(54.7 ) cos(45 ) = 0.45 MPa (65.1 psi)
37 7.15 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 the (111) plane in each of the [11 0], [101 ] and [011 ] directions. In order to solve this problem it is necessary to employ Equation 7.4, but first we need to solve for the for λ and φ angles for the three slip systems. For each of these three slip systems, the φ will be the same i.e., the angle between the direction of the applied stress, [110] and the normal to the (111) plane, that is, the [111] direction. The angle φ may be determined using Equation 7.6 as φ = cos 1 u 1 u + v 1 v + w 1 w u 1 + v1 + w1 ( ) u + v + w ( ) where (for [110]) u 1 = 1, v 1 = 1, w 1 = 0, and (for [111]) u = 1, v = 1, w = 1. Therefore, φ is equal to φ = cos 1 (1)(1) + (1)(1) + (0)(1) (1) + (1) + (0) [ ][(1) + (1) + (1) ] = cos 1 6 = 35.3 Let us now determine λ for the [11 0 ] slip direction. Again, using Equation 7.6 where u 1 = 1, v 1 = 1, w 1 = 0 (for [110]), and u = 1, v = 1, w = 0 (for [11 0]. Therefore, λ is determined as λ [110] [11 0] = cos 1 (1)(1) + (1)( 1) + (0)(0) (1) + (1) + (0) [ ][(1) + ( 1) + (0) ] = cos 1 0 = 90 Now, we solve for the yield strength for this (111) [110] slip system using Equation 7.4 as
38 σ y = τ crss (cosφ cos λ) = 1.75 MPa 1.75 MPa = cos (35.3 ) cos (90 ) (0.816) (0) = which means that slip will not occur on this (111) [110] slip system. Now, we must determine the value of λ for the (111) [101] slip system that is, the angle between the [110] and [101] directions. Again using Equation 7.6 λ [110] [101] = cos 1 (1)(1) + (1)(0) + (0)( 1) [(1) + (1) + (0) ][(1) + (0) + ( 1) ] = cos 1 1 = 60 Now, we solve for the yield strength for this (111) [101] slip system using Equation 7.4 as σ y = τ crss (cosφ cos λ) = 1.75 MPa cos (35.3 ) cos (60 ) = 1.75 MPa = 4.9 MPa (0.816) (0.500) And, finally, for the (111) [011] slip system, λ is computed using Equation 7.6 as follows: λ [110] [011] = cos 1 (1)(0) + (1)(1) + (0)( 1) [(1) + (1) + (0) ][(0) + (1) + ( 1) ] = cos 1 1 = 60 Thus, since the values of φ and λ for this (110) [011 ] slip system are the same as for (111) [101 ], so also will σ y be the same viz 4.9 MPa.
39 7.16 (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.75 MPa, compute the resolved shear stress in the [1 11] direction on each of the (110) and (101) planes. (b) On the basis of these resolved shear stress values, which slip system(s) is (are) most favorably oriented? (a) This part of the problem asks, for a BCC metal, that we compute the resolved shear stress in the [111] direction on each of the (110) and (101) planes. In order to solve this problem it is necessary to employ Equation 7., which means that we first need to solve for the for angles λ and φ for the three slip systems. For each of these three slip systems, the λ will be the same i.e., the angle between the direction of the applied stress, [010] and the slip direction, [111]. This angle λ may be determined using Equation 7.6 λ = cos 1 u 1 u + v 1 v + w 1 w u 1 + v1 + w1 ( ) u + v + w ( ) where (for [010]) u 1 = 0, v 1 = 1, w 1 = 0, and (for [1 11]) u = 1, v = 1, w = 1. Therefore, λ is determined as λ = cos 1 (0)( 1) + (1)(1) + (0)(1) (0) + (1) + (0) [ ][( 1) + (1) + (1) ] = cos = 54.7 Let us now determine φ for the angle between the direction of the applied tensile stress i.e., the [010] direction and the normal to the (110) slip plane i.e., the [110] direction. Again, using Equation 7.6 where u = 0, v = 1, w = 0 (for [010]), and u = 1, v = 1, w = 0 (for [110]), φ is equal to φ [010] [110] = cos 1 (0)(1) + (1)(1) + (0)(0) (0) + (1) + (0) [ ][(1) + (1) + (0) ] = cos 1 1 = 45
40 Now, using Equation 7. τ R = σ cosφ cos λ we solve for the resolved shear stress for this slip system as τ R (110) [1 11] = (.75 MPa) [ cos (54.7 ) cos (45 )] = (.75 MPa) (0.578)(0.707) = 1.1 MPa Now, we must determine the value of φ for the (101) [111] slip system that is, the angle between the direction of the applied stress, [010], and the normal to the (101) plane i.e., the [101] direction. Again using Equation 7.6 λ [010] [101] = cos 1 (0)(1) + (1)(0) + (0)(1) (0) + (1) + (0) [ ][(1) + (0) + (1) ] = cos 1 (0) = 90 Thus, the resolved shear stress for this (101) [111] slip system is τ R (101) [1 11] = = (.75 MPa) [ cos (54.7 ) cos (90 )] = (.75 MPa) (0.578)(0) = 0 MPa (b) The most favored slip system(s) is (are) the one(s) that has (have) the largest τ R value. Therefore, the (110) [1 11] is the most favored since its τ R (1.1 MPa) is greater than the τ R value for (101) [1 11] (viz., 0 MPa).
41 7.17 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 [1 0] 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 shear stress is 3.4 MPa. This problem asks for us to determine the tensile stress at which a FCC metal yields when the stress is applied along a [10] direction such that slip occurs on a (111) plane and in a [101] direction; the critical resolved shear stress for this metal is 3.4 MPa. To solve this problem we use Equation 7.4; however it is first necessary to determine the values of φ and λ. These determinations are possible using Equation 7.6. Now, λ is the angle between [10] and [101] directions. Therefore, relative to Equation 7.6 let us take u 1 = 1, v 1 = 0, and w 1 =, as well as u = 1, v = 0, and w = 1. This leads to λ = cos 1 u 1 u + v 1 v + w 1 w u 1 + v1 + w1 ( ) u + v + w ( ) = cos 1 ( 1)( 1) + (0)(0) + ()(1) ( 1) + (0) + () [ ][( 1) + (0) + (1) ] = cos = 18.4 Now for the determination of φ, the normal to the (111) slip plane is the [111] direction. Again using Equation 7.6, where we now take u 1 = 1, v 1 = 0, w 1 = (for [1 0]), and u = 1, v = 1, w = 1 (for [111]). Thus, φ = cos 1 ( 1)(1) + (0)(1) + ()(1) ( 1) + (0) + () [ ][(1) + (1) + (1) ] = cos = 39. It is now possible to compute the yield stress (using Equation 7.4) as
42 σ y = τ crss cosφ cos λ = 3.4 MPa = 4.65 MPa
43 7.18 The critical resolved shear stress for iron is 7 MPa (4000 psi). Determine the maximum possible yield strength for a single crystal of Fe pulled in tension. In order to determine the maximum possible yield strength for a single crystal of Fe pulled in tension, we simply employ Equation 7.5 as σ y = τ crss = ()(7 MPa) = 54 MPa (8000 psi)
44 Deformation by Twinning 7.19 List four major differences between deformation by twinning and deformation by slip relative to mechanism, conditions of occurrence, and final result. Four major differences between deformation by twinning and deformation by slip are as follows: (1) with slip deformation there is no crystallographic reorientation, whereas with twinning there is a reorientation; () for slip, the atomic displacements occur in atomic spacing multiples, whereas for twinning, these displacements may be other than by atomic spacing multiples; (3) slip occurs in metals having many slip systems, whereas twinning occurs in metals having relatively few slip systems; and (4) normally slip results in relatively large deformations, whereas only small deformations result for twinning.
45 Strengthening by Grain Size Reduction 7.0 Briefly explain why smallangle grain boundaries are not as effective in interfering with the slip process as are highangle grain boundaries. Smallangle grain boundaries are not as effective in interfering with the slip process as are highangle grain boundaries because there is not as much crystallographic misalignment in the grain boundary region for smallangle, and therefore not as much change in slip direction.
46 7.1 Briefly explain why HCP metals are typically more brittle than FCC and BCC metals. Hexagonal close packed metals are typically more brittle than FCC and BCC metals because there are fewer slip systems in HCP.
47 7. Describe in your own words the three strengthening mechanisms discussed in this chapter (i.e., grain size reduction, solidsolution strengthening, and strain hardening). Be sure to explain how dislocations are involved in each of the strengthening techniques. These three strengthening mechanisms are described in Sections 7.8, 7.9, and 7.10.
48 7.3 (a) From the plot of yield strength versus (grain diameter) 1/ for a 70 Cu 30 Zn cartridge brass, Figure 7.15, determine values for the constants σ 0 and k y in Equation 7.7. (b) Now predict the yield strength of this alloy when the average grain diameter is mm. (a) Perhaps the easiest way to solve for σ 0 and k y in Equation 7.7 is to pick two values each of σ y and d 1/ from Figure 7.15, and then solve two simultaneous equations, which may be created. For example d 1/ (mm) 1/ σ y (MPa) The two equations are thus 75 = σ k y 175 = σ k y of these equations yield the values of k y = 1.5 MPa (mm) 1/ [ 1810 psi(mm) 1/ ] σ 0 = 5 MPa (3630 psi) (b) When d = mm, d 1/ = 31.6 mm 1/, and, using Equation 7.7, σ y = σ 0 + k y d 1/ = (5 MPa) MPa (mm) 1/ (31.6 mm1/ ) = 40 MPa (61,000 psi)
49 7.4 The lower yield point for an iron that has an average grain diameter of mm is 135 MPa (19,500 psi). At a grain diameter of mm, the yield point increases to 60 MPa (37,500 psi). At what grain diameter will the lower yield point be 05 MPa (30,000 psi)? The best way to solve this problem is to first establish two simultaneous expressions of Equation 7.7, solve for σ 0 and k y, and finally determine the value of d when σ y = 05 MPa. The data pertaining to this problem may be tabulated as follows: σ y d (mm) d 1/ (mm) 1/ 135 MPa MPa The two equations thus become 135 MPa = σ 0 + (4.47) k y 60 MPa = σ 0 + (11.18) k y Which yield the values, σ 0 = 51.7 MPa and k y = MPa(mm) 1/. At a yield strength of 05 MPa 05 MPa = 51.7 MPa + [ MPa (mm) 1/ ]d 1/ or d 1/ = 8.3 (mm) 1/, which gives d = mm.
50 7.5 If it is assumed that the plot in Figure 7.15 is for noncoldworked brass, determine the grain size of the alloy in Figure 7.19; assume its composition is the same as the alloy in Figure This problem asks that we determine the grain size of the brass for which is the subject of Figure From Figure 7.19a, the yield strength of brass at 0%CW is approximately 175 MPa (6,000 psi). This yield strength from Figure 7.15 corresponds to a d 1/ value of approximately 1.0 (mm) 1/. Thus, d = mm.
51 Solid Strengthening 7.6 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. Below is shown an edge dislocation and where an interstitial impurity atom would be located. Compressive lattice strains are introduced by the impurity atom. There will be a net reduction in lattice strain energy when these lattice strains partially cancel tensile strains associated with the edge dislocation; such tensile strains exist just below the bottom of the extra halfplane of atoms (Figure 7.4).
52 Strain Hardening 7.7 (a) Show, for a tensile test, that ε %CW = 100 ε+1 if there is no change in specimen volume during the deformation process (i.e., A 0 l 0 = A d l d ). (b) Using the result of part (a), compute the percent cold work experienced by naval brass (the stressstrain behavior of which is shown in Figure 6.1) when a stress of 400 MPa (58,000 psi) is applied. (a) From Equation 7.8 %CW = A 0 A d A = 1 A d 100 A 0 Which is also equal to 1 l l d since A d /A 0 = l 0 /l d, the conservation of volume stipulation given in the problem statement. Now, from the definition of engineering strain (Equation 6.) ε = l d l 0 l 0 = l d l 0 1 Or, l 0 l d = 1 ε + 1 Substitution for l 0 / l d into the %CW expression above gives %CW = 1 l = 1 l d 1 ε 100 = 100 ε + 1 ε + 1
53 expression (b) From Figure 6.1, a stress of 400 MPa (58,000 psi) corresponds to a strain of Using the above ε 0.13 %CW = 100 = 100 = 11.5%CW ε
54 7.8 Two previously undeformed cylindrical specimens of an alloy are to be strain hardened by reducing their crosssectional 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 radius of 1 mm, must have the same deformed hardness as the first specimen; compute the second specimen's radius after deformation. In order for these two cylindrical specimens to have the same deformed hardness, they must be deformed to the same percent cold work. For the first specimen %CW = A 0 A d 100 = π r 0 π r d A 0 π r = π (16 mm) π (11 mm) π (16 mm) 100 = 5.7%CW For the second specimen, the deformed radius is computed using the above equation and solving for r d as r d = r 0 1 %CW 100 = (1 mm) 1 5.7%CW 100 = 8.5 mm
55 7.9 Two previously undeformed specimens of the same metal are to be plastically deformed by reducing their crosssectional 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 such. Their original and deformed dimensions are as follows: Circular (diameter, mm) Rectangular (mm) Original dimensions Deformed dimensions Which of these specimens will be the hardest after plastic deformation, and why? The hardest specimen will be the one that has experienced the greatest degree of cold work. Therefore, all we need do is to compute the %CW for each specimen using Equation 7.8. For the circular one %CW = A 0 A d 100 A 0 = π r 0 π r d π r mm 11.4 mm π π = 15. mm π 100 = 43.8%CW For the rectangular one (15 mm)(175 mm) (75 mm)(00 mm) %CW = 100 = 31.4%CW (15 mm)(175 mm) Therefore, the deformed circular specimen will be harder.
56 7.30 A cylindrical specimen of coldworked copper has a ductility (%EL) of 5%. If its coldworked radius is 10 mm (0.40 in.), what was its radius before deformation? This problem calls for us to calculate the precoldworked radius of a cylindrical specimen of copper that has a coldworked ductility of 5%EL. From Figure 7.19c, copper that has a ductility of 5%EL will have experienced a deformation of about 11%CW. For a cylindrical specimen, Equation 7.8 becomes %CW = π r 0 π r d π r Since r d = 10 mm (0.40 in.), solving for r 0 yields r 0 = r d 1 %CW 100 = 10 mm = 10.6 mm (0.44 in.) 100
57 psi)? psi)? 7.31 (a) What is the approximate ductility (%EL) of a brass that has a yield strength of 75 MPa (40,000 (b) What is the approximate Brinell hardness of a 1040 steel having a yield strength of 690 MPa (100,000 (a) In order to solve this problem, it is necessary to consult Figures 7.19a and 7.19c. From Figure 7.19a, a yield strength of 75 MPa for brass corresponds to 10%CW. A brass that has been coldworked 10% will have a ductility of about 43%EL [Figure 7.19c]. (b) This portion of the problem asks for the Brinell hardness of a 1040 steel having a yield strength of 690 MPa (100,000 psi). From Figure 7.19a, a yield strength of 690 MPa for a 1040 steel corresponds to about 10%CW. A 1040 steel that has been cold worked 10% will have a tensile strength of about 780 MPa [Figure 7.19b]. Finally, using Equation 6.0a HB = TS (MPa) 3.45 = 780 MPa 3.45 = 6
58 7.3 Experimentally, it has been observed for single crystals of a number of metals that the critical resolved shear stress τ crss is a function of the dislocation density ρ D as τ crss = τ 0 + A ρ D where τ 0 and A are constants. For copper, the critical resolved shear stress is.10 MPa (305 psi) at a dislocation density of 10 5 mm . If it is known that the value of A for copper is MPamm (0.9 psimm), compute the τ crss at a dislocation density of 10 7 mm . We are asked in this problem to compute the critical resolved shear stress at a dislocation density of 10 7 mm . It is first necessary to compute the value of the constant τ 0 (in the equation provided in the problem statement) from the one set of data as τ 0 = τ crss A ρ D =.10 MPa ( MPa  mm) ( 10 5 mm ) = 0.09 MPa (13.3 psi) Now, the critical resolved shear stress may be determined at a dislocation density of 10 7 mm  as τ crss = τ 0 + A ρ D = (0.09 MPa) + ( MPa  mm) 10 7 mm = 0. MPa (90 psi)
59 Recovery Recrystallization Grain Growth 7.33 Briefly cite the differences between recovery and recrystallization processes. For recovery, there is some relief of internal strain energy by dislocation motion; however, there are virtually no changes in either the grain structure or mechanical characteristics. During recrystallization, on the other hand, a new set of strainfree grains forms, and the material becomes softer and more ductile.
60 7.34 Estimate the fraction of recrystallization from the photomicrograph in Figure 7.1c. Below is shown a square grid onto which is superimposed the recrystallized regions from the micrograph. Approximately 400 squares lie within the recrystallized areas, and since there are 67 total squares, the specimen is about 60% recrystallized.
61 7.35 Explain the differences in grain structure for a metal that has been cold worked and one that has been cold worked and then recrystallized. During coldworking, the grain structure of the metal has been distorted to accommodate the deformation. Recrystallization produces grains that are equiaxed and smaller than the parent grains.
62 7.36 (a) What is the driving force for recrystallization? (b) For grain growth? (a) The driving force for recrystallization is the difference in internal energy between the strained and unstrained material. (b) The driving force for grain growth is the reduction in grain boundary energy as the total grain boundary area decreases.
63 7.37 (a) From Figure 7.5, 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. (a) At 500 C, the time necessary for the average grain diameter to grow to increase from 0.01 to 0.1 mm is approximately 3500 min. (b) At 600 C the time required for this same grain size increase is approximately 150 min.
64 7.38 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: Time (min) Grain Diameter (mm) (a) What was the original grain diameter? (b) What grain diameter would you predict after 150 min at 650 C? (a) Using the data given and Equation 7.9 (taking n = ), we may set up two simultaneous equations with d 0 and K as unknowns; thus ( mm) d 0 = (30 min) K ( mm) d 0 = (90 min) K of these expressions yields a value for d 0, the original grain diameter, of and a value for K of mm /min d 0 = 0.01 mm, (b) At 150 min, the diameter d is computed using a rearranged form of Equation 7.9 as d = d 0 + Kt = (0.01 mm) + ( mm /min)(150 min) = mm
65 7.39 An undeformed specimen of some alloy has an average grain diameter of mm. You are asked to reduce its average grain diameter to mm. Is this possible? If so, explain the procedures you would use and name the processes involved. If it is not possible, explain why. Yes, it is possible to reduce the average grain diameter of an undeformed alloy specimen from mm to mm. In order to do this, plastically deform the material at room temperature (i.e., cold work it), and then anneal at an elevated temperature in order to allow recrystallization and some grain growth to occur until the average grain diameter is mm.
66 7.40 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 of your intuition, cite an explicit expression for this temperature dependence. (a) The temperature dependence of grain growth is incorporated into the constant K in Equation 7.9. (b) The explicit expression for this temperature dependence is of the form K = K 0 exp Q RT in which K 0 is a temperatureindependent constant, the parameter Q is an activation energy, and R and T are the gas constant and absolute temperature, respectively.
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