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CHAPTER 8
DEFORMATION AND STRENGTHENING MECHANISMS PROBLEM SOLUTIONS
Basic Concepts of Dislocations Characteristics of Dislocations
8.1 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 mm3 of material having a density of 105 mm-2 is just
(105
mm -2 )(1000 mm 3 ) = 108 mm = 105 m = 62 mi
Similarly, for a dislocation density of 109 mm-2, the total length is
(109 mm -2 )(1000
mm 3 ) = 1012 mm = 109 m = 6.2 x 105 mi
8.2 When the two edge dislocations become aligned, a planar region of vacancies will exist between the dislocations as:
8.3 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.
8.4 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 dislocation--parallel screw dislocation--perpendicular mixed dislocation--neither parallel nor perpendicular
Slip Systems
8.5 (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.
8.6 (a) For the FCC crystal structure, the planar density for the (110) plane is given in Equation 3.13 as
1 4R2 2 0.177 R2
PD110 (FCC) =
=
Furthermore, the planar densities of the (100) and (111) planes are calculated in Homework Problem 3.75, which are as follows:
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PD100 (FCC) =
1 4R2 1
=
0.25 R2 0.29 R2
PD111 (FCC) =
2R2
3
=
(b) For the BCC crystal structure, the planar densities of the (100) and (110) planes were determined in Homework Problem 3.76, which are as follows: 3 16R 2 3 8R2 2 0.19 R2 0.27 R2
PD100 (BCC) =
=
PD110 (BCC) =
=
Below is a BCC unit cell, within which is shown a (111) plane.
(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.
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(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 one-half atom. In Figure (b) the triangle with A, B, and C at its corners is an equilateral triangle. And, from Figure (b), xy . The triangle edge length, x, is equal to the length of a face diagonal, as indicated in the area of this triangle is 2
Figure (a). And its length is related to the unit cell edge length, a, as
x 2 = a 2 + a 2 = 2a 2
or x= a
4R 3
2
For BCC, a =
(Equation 3.3), and, therefore,
x=
4R 2 3
Also, from Figure (b), with respect to the length y we may write x y2 + 2
3 2
2
= x2
which leads to y =
x
. And, substitution for the above expression for x yields
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y=
x 2
3
4R 2 3 4R 2 = = 2 2 3
Thus, the area of this triangle is equal to 1 4R 2 4R 2 8R2 1 = xy = 2 2 3 2 3
AREA =
And, finally, the planar density for this (111) plane is
0.5 atom 8R2 3 3 16 R 2 0.11 R2
PD111 (BCC) =
=
=
8.7 Below is shown the atomic packing for a BCC {110} type plane. The arrows indicate two different <111> type directions.
8.8 Below is shown the atomic packing for an HCP {0001} type plane. The arrows indicate three different < 1120 > type directions.
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Slip in Single Crystals
8.9 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 [1 1 0] direction as noted in the
figure below.
The angle between the [100] and [1 1 0] directions, , may be determined using Equation 8.5
= cos -1
(
2 2 2 2 2 2 u1 + v1 + w1 u2 + v2 + w 2
u1u 2 + v1v 2 + w1w 2
)(
)
where (for [100]) u1 = 1, v1 = 0, w1 = 0, and (for [1 1 0]) u2 = 1, v2 = 1, w2 = 0. Therefore, is equal to
= cos -1
[
(1) 2 + (0) 2 + (0) 2 (1)2 + ( -1) 2 + (0) 2
(1)(1) + (0)(-1) + (0)(0)
][
]
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1 = 45 = cos-1 2
Now, the angle is equal to the angle between the normal to the (111) plane (which is the [111] direction), and the [100] direction. Again from Equation 8.5, and for u1 = 1, v1 = 1, w1 = 1, u2 = 1, v2 = 0, and w2 = 0, we have
= cos -1
[(1)2 + (1)2 + (1) 2 ][
(1)(1) + (1)( 0) + (1)(0)
(1) 2 + (0) 2 + (0) 2
]
1 = 54.7 = cos-1 3
Therefore, the Schmid factor is equal to
1 1 = 0.408 cos cos = cos(45) cos(54.7) = 2 3
8.10 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 = 60, = 35, and that the values of the critical resolved shear stress and applied tensile stress are 6.2 MPa (900 psi) and 12 MPa (1750 psi), respectively. From Equation 8.1
R = cos cos = (12 MPa)(cos 60 )(cos 35 ) = 4.91 MPa (717 psi)
Since the resolved shear stress (4.91 MPa) is less that the critical resolved shear stress (6.2 MPa), the single crystal will not yield. However, from Equation 8.3, the stress at which yielding occurs is
y =
crss cos cos
=
6.2 MPa = 15.1 MPa (2200 psi ) (cos 60)(cos 35)
8.11 We are asked to compute the critical resolved shear stress for Zn. As stipulated in the problem, = 65, while possible values for are 30, 48, and 78.
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(a) Slip will occur along that direction for which (cos cos ) is a maximum, or, in this case, for the largest cos . The cosines for the possible values are given below. cos(30) = 0.87 cos(48) = 0.67 cos(78) = 0.21 Thus, the slip direction is at an angle of 30 with the tensile axis. (b) From Equation 8.3, the critical resolved shear stress is just crss = y (cos cos ) max = (2.5 MPa) cos(65) cos(30) = 0.90 MPa (130 psi)
[
]
8.12 This problem asks that we compute the critical resolved shear stress for silver. In order to do this, we must employ Equation 8.3, but first it is necessary to solve for the angles and which are shown in the sketch below.
The angle is the angle between the tensile axis--i.e., along the [001] direction--and the slip direction--i.e., [1 01]. The angle may be determined using Equation 8.5 as = cos -1 2 2 2 2 2 2 u1 + v1 + w1 u2 + v2 + w 2
u1u 2 + v1v 2 + w1w 2
(
)(
)
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where (for [001]) u1 = 0, v1 = 0, w1 = 1, and (for [1 01]) u2 = 1, v2 = 0, w2 = 1. Therefore, is equal to
= cos -1
[
(0) 2 + (0) 2 + (1) 2 (-1) 2 + (0) 2 + (1) 2 (0)(-1) + (0)(0) + (1)(1)
][
]
1 = 45 = cos-1 2
Furthermore, 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 8.5 (1) 2 + (1) 2 + (1) 2
= cos -1
[(0)2 + (0)2 + (1)2 ][
(0)(1) + (0)(1) + (1)(1)
]
1 = 54.7 = cos-1 3
And, finally, using Equation 8.3, the critical resolved shear stress is equal to crss = y (cos cos ) 1 1 = (1.1 MPa) [cos(54.7) cos(45)] = (1.1 MPa) = 0.45 MPa (65.1 psi) 3 2
8.13 This problem asks that, for a metal that has the FCC crystal structure, we compute the applied stress(s) that are required to cause slip to occur on a (111) plane in each of the [1 1 0 ], [10 1 ], and [0 1 1 ] directions. In order to solve this problem it is necessary to employ Equation 8.3, but first we need to solve for the for and angles for the three slip systems. For each of these three slip systems, the will be the same--i.e., the angle between the direction of the applied stress, [100] and the normal to the (111) plane, that is, the [111] direction. The angle may be determined using Equation 8.5 as
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= cos -1
(
2 2 2 2 2 u1 + v1 + w12 u 2 + v 2 + w 2
u1u 2 + v1v2 + w1w 2
)(
)
where (for [100]) u1 = 1, v1 = 0, w1 = 0, and (for [111]) u2 = 1, v2 = 1, w2 = 1. Therefore, is equal to = cos -1 (1) 2 + (1) 2 + (1) 2
[(1)2 + (0)2 + (0)2 ][
(1)(1) + (0)(1) + (0)(1)
]
1 = 54.7 = cos-1 3 Let us now determine for the [1 1 0 ] slip direction. Again, using Equation 8.5 where u1 = 1, v1 = 0, w1 = 0 (for [100]), and u2 = 1, v2 = 1, w2 = 0 (for [11 0]. Therefore, is determined as = cos -1 [100 ]-[11 0] (1) 2 + (0) 2 + (0) 2 (1) 2 + (-1) 2 + (0) 2 (1)(1) + (0)(-1) + (0)(0)
[
][
]
1 = 45 = cos-1 2
Now, we solve for the yield strength for this (111) [1 1 0] slip system using Equation 8.3 as
crss
y =
(cos cos)
=
0.5 MPa 0.5 MPa = = 1.22 MPa cos(54.7 ) cos(45) (0.578) (0.707)
Now, we must determine the value of for the (111) [101 ] slip system--that is, the angle between the [100] and [101 ] directions. Again using Equation 8.5
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= cos -1 [100 ]-[10 1 ]
[
(1) 2 + (0) 2 + (0) 2 (1) 2 + (0) 2 + (-1) 2 (1)(1) + (0)(0) + (0)(-1)
][
]
1 = 45 = cos-1 2 Thus, since the values of and for this (111) [101 ] slip system are the same as for (111) [1 1 0], so also will y be the same--viz 1.22 MPa. And, finally, for the (111) [0 1 1] slip system, is computed using Equation 8.5 as follows: (1) 2 + (0) 2 + (0) 2 (0) 2 + (-1) 2 + (1) 2 (1)(0) + (0)(-1) + (0)(1)
[100 ]-[ 01 1]
= cos -1
[
][
]
= cos-1 (0) = 90
Thus, from Equation 8.3, the yield strength for this slip system is crss
y =
(cos cos)
=
0.5 MPa 0.5 MPa = = cos(54.7 ) cos(90) (0.578) (0)
which means that slip will not occur on this (111) [0 1 1 ] slip system.
8.14 (a) This part of the problem asks, for a BCC metal, that we compute the resolved shear stress in the [1 1 1 ] direction on each of the (110), (011), and (101 ) planes. In order to solve this problem it is necessary to employ Equation 8.1, which means that we first need to solve for the for angles and for the three slip systems. For each of these three slip systems, the will be the same--i.e., the angle between the direction of the applied stress, [100] and the slip direction, [1 1 1]. This angle may be determined using Equation 8.5 = cos -1 2 2 2 2 2 2 u1 + v1 + w1 u2 + v2 + w 2
u1u 2 + v1v 2 + w1w 2
(
)(
)
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where (for [100]) u1 = 1, v1 = 0, w1 = 0, and (for [1 1 1]) u2 = 1, v2 = 1, w2 = 1. Therefore, is determined as = cos -1 (1) 2 + (0) 2 + (0) 2 (1)2 + (-1) 2 + (1) 2 (1)(1) + (0)(-1) + (0)(1)
[
][
]
1 = 54.7 = cos-1 3
Let us now determine for the angle between the direction of the applied tensile stress--i.e., the [100] direction-- and the normal to the (110) slip plane--i.e., the [110] direction. Again, using Equation 8.5 where u1 = 1, v1 = 0, w1 = 0 (for [100]), and u2 = 1, v2 = 1, w2 = 0 (for [110]), is equal to [100 ]-[110] = cos -1 (1) 2 + (0) 2 + (0) 2 (1)2 + (1)2 + (0)2 (1)(1) + (0)(1) + (0)( 0)
[
][
]
1 = 45 = cos-1 2
Now, using Equation 8.1 R = cos cos we solve for the resolved shear stress for this slip system as
R (110) -[11 1] = (4.0 MPa ) cos(45) cos(54.7) = (4.0 MPa ) (0.707)(0.578) = 1.63 MPa
[
]
Now, we must determine the value of for the (011) [1 1 1] slip system--that is, the angle between the direction of the applied stress, [100], and the normal to the (011) plane--i.e., the [011] direction. Again using Equation 8.5 [100 ]-[ 011] = cos -1 (1) 2 + (0) 2 + (0) 2 (0) 2 + (1) 2 + (1) 2 (1)(0) + (0)(1) + (0)(1)
[
][
]
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= cos-1 (0) = 90 Thus, the resolved shear stress for this (011) [1 1 1] slip system is R (011) -[11 1] = = (4.0 MPa) cos(90) cos (54.7) = (4.0 MPa) (0)( 0.578) = 0 MPa
[
]
And, finally, it is necessary to determine the value of for the (10 1 ) [1 1 1 ] slip system --that is, the angle between the direction of the applied stress, [100], and the normal to the (10 1 ) plane--i.e., the [10 1 ] direction. Again using Equation 8.5 = cos -1 [100 ]-[10 1 ] (1) 2 + (0) 2 + (0) 2 (1) 2 + (0) 2 + (-1) 2 (1)(1) + (0)(0) + (0)(-1)
[
][
]
1 = 45 = cos-1 2 Here, as with the the (110) [1 1 1 ] slip system above, the value of is 45, which again leads to R (10 1 ) -[11 1] = (4.0 MPa) cos(45 ) cos(54.7 ) = (4.0 MPa) (0.707)(0.578) = 1.63 MPa (b) The most favored slip system(s) is (are) the one(s) that has (have) the largest R value. Both (110) [1 1 1 ] and (10 1 ) - [1 1 1 ] slip systems are most favored since they have the same R (1.63 MPa), which is greater than the R value for (011) - [1 1 1 ] (viz., 0 MPa).
[
]
8.15 This problem asks for us to determine the tensile stress at which a BCC metal yields when the stress is applied along a [121] direction such that slip occurs on a (101) plane and in a [1 11] direction; the critical resolved shear stress for this metal is 2.4 MPa. To solve this problem we use Equation 8.3; however it is first necessary to determine the values of and . These determinations are possible using Equation 8.5. Now, is the angle between [121] and [1 11 ] directions. Therefore, relative to Equation 8.5 let us take u1 = 1, v1 = 2, and w1 = 1, as well as u2 = 1, v2 = 1, and w2 = 1. This leads to
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= cos -1 = cos-1
(
2 2 2 2 2 2 u1 + v1 + w1 u2 + v2 + w 2
u1u 2 + v1v 2 + w1w 2
)(
)
[
2 + (2) 2 + (1) 2 ( -1)2 + (1) 2 + (1) 2 (1)
(1)( - 1) + (2)(1) + (1)(1)
][
]
2 = 61.9 = cos-1 18
Now for the determination of , the normal to the (101) slip plane is the [101] direction. Again using Equation 8.5, where we now take u1 = 1, v1 = 2, w1 = 1 (for [121]), and u2 = 1, v2 = 0, w2 = 1 (for [101]). Thus,
= cos -1 2 + (2) 2 + (1) 2 (1) 2 + (0) 2 + (1)2 (1)
[
(1)(1) + (2)(0) + (1)(1)
][
]
2 = 54.7 = cos-1 12
It is now possible to compute the yield stress (using Equation 8.3) as
crss
y =
cos cos
=
2.4 MPa = 8.82 MPa 2 2 12 18
8.16 In order to determine the maximum possible yield strength for a single crystal of Cu pulled in tension, we simply employ Equation 8.4 as
y = 2 crss = (2)(0.48 MPa) = 0.96 MPa (140 psi)
Deformation by Twinning
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8.17 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; (2) 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.
Strengthening by Grain Size Reduction
8.18 Small-angle grain boundaries are not as effective in interfering with the slip process as are high-angle 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.
8.19 Hexagonal close packed metals are typically more brittle than FCC and BCC metals because there are fewer slip systems in HCP.
8.20 These three strengthening mechanisms are described in Sections 8.9, 8.10, and 8.11.
8.21 (a) Perhaps the easiest way to solve for 0 and ky in Equation 8.6 is to pick two values each of y and d-1/2 from Figure 8.15, and then solve two simultaneous equations, which may be set up. For example d-1/2 (mm) -1/2 4 12
y (MPa)
75 175
The two equations are thus 75 = 0 + 4 k y 175 = 0 + 12 k y
These yield the values of
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k y = 12.5 MPa (mm)1/2 1810 psi (mm)1/2 0 = 25 MPa (3630 psi) (b) When d = 1.0 x 10-3 mm, d-1/2 = 31.6 mm-1/2, and, using Equation 8.6,
y = 0 + k yd
-1/2
1/2 = (25 MPa) + 12.5 MPa( mm) ( 31.6 mm -1/2 ) = 420 MPa (61, 000 psi)
8.22 We are asked to determine the grain diameter for an iron which will give a yield strength of 205 MPa (30,000 psi). The best way to solve this problem is to first establish two simultaneous expressions of Equation 8.6, solve for 0 and ky, and finally determine the value of d when y = 205 MPa. The data pertaining to this problem may be tabulated as follows: y 135 MPa 260 MPa The two equations thus become 135 MPa = 0 + (4.47) k y 260 MPa = 0 + (11.18) k y Which yield the values, 0 = 51.7 MPa and ky = 18.63 MPa(mm)1/2. At a yield strength of 205 MPa 205 MPa = 51.7 MPa + 18.63 MPa (mm)1/2 d -1/2 or d-1/2 = 8.23 (mm)-1/2, which gives d = 1.48 x 10-2 mm. d (mm) 5 x 10-2 8 x 10
-3
d-1/2 (mm)-1/2 4.47 11.18
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8.23 This problem asks that we determine the grain size of the brass for which is the subject of Figure 8.19. From Figure 8.19(a), the yield strength of brass at 0%CW is approximately 175 MPa (26,000 psi). This yield strength from Figure 8.15 corresponds to a d-1/2 value of approximately 12.0 (mm)-1/2. Thus, d = 6.9 x 10-3 mm.
Solid-Solution Strengthening
8.24 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 half-plane of atoms (Figure 8.4).
Strain Hardening
8.25 (a) We are asked to show, for a tensile test, that %CW = x 100 + 1
From Equation 8.7 A - Ad %CW = 0 A0 A x 100 = 1 - d x 100 A0
Which is also equal to l 1 - 0 x 100 ld
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since Ad/A0 = l0/ld, the conservation of volume stipulation given in the problem statement. Now, from the definition of engineering strain (Equation 7.2) ld ld - l0 = -1 l l
0 0
=
Or, l0 l
d
=
1 +1
Substitution in the %CW expression above gives l %CW = 1 - 0 ld 1 x 100 = 1 - x 100 = x 100 + 1 + 1
(b) From Figure 7.12, a stress of 415 MPa (60,000 psi) corresponds to a strain of 0.16. Using the above expression 0.16 %CW = x 100 = 0.16 + 1.00 x 100 = 13.8%CW + 1
8.26 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
A0 - Ad A
0 2 2 r0 - rd 2 r0
%CW =
x 100 =
x 100
=
(15 mm ) 2 - (12 mm ) 2 (15 mm) 2
x 100 = 36%CW
For the second specimen, the deformed radius is computed using the above equation and solving for rd as %CW 100
rd = r0 1 -
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= (11 mm) 1 -
36% CW = 8.80 mm 100
8.27 We are given the original and deformed cross-sectional dimensions for two specimens of the same metal, and are then asked to determine which is the hardest after deformation. 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 8.7. For the circular one A - Ad x 100 %CW = 0 A0 r 2 - r 2 d x 100 = 0 2 r 0
2 2 15.2 mm - 11.4 mm 2 2 = x 100 = 43.8%CW 15.2 mm 2 2
For the rectangular one (125 mm)(175 mm ) - (75 mm )(200 mm ) %CW = x 100 = 31.4%CW (125 mm )(175 mm )
Therefore, the deformed circular specimen will be harder.
8.28 This problem calls for us to calculate the precold-worked radius of a cylindrical specimen of copper that has a cold-worked ductility of 25%EL. From Figure 8.19(c), copper that has a ductility of 25%EL will have experienced a deformation of about 11%CW. For a cylindrical specimen, Equation 8.7 becomes
2 2 r 0 - r d x 100 %CW = r 2 0
Since rd = 10 mm (0.40 in.), solving for r0 yields
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r0 =
rd % CW 1 - 100
=
10 mm 11.0 1 - 100
= 10.6 mm (0.424 in.)
8.29 (a) We want to compute the ductility of a brass that has a yield strength of 275 MPa (40,000 psi). In order to solve this problem, it is necessary to consult Figures 8.19(a) and (c). From Figure 8.19(a), a yield strength of 275 MPa for brass corresponds to 10%CW. A brass that has been cold-worked 10% will have a ductility of about 44%EL [Figure 8.19(c)]. (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 8.19(a), a yield strength of 690 MPa for a 1040 steel corresponds to about 11%CW. A 1040 steel that has been cold worked 11% will have a tensile strength of about 790 MPa [Figure 8.19(b)]. Finally, using Equation 7.25a HB = TS (MPa) 790 MPa = = 230 3.45 3.45
8.30 We are asked in this problem to compute the critical resolved shear stress at a dislocation density of 10 mm . It is first necessary to compute the value of the constant A (in the equation provided in the problem statement) from the one set of data as crss - 0 D 0.69 MPa - 0.069 MPa 10
4
6 -2
A=
=
mm
-2
= 6.21 x 10
-3
MPa - mm (0.90 psi - mm)
Now, the critical resolved shear stress may be determined at a dislocation density of 106 mm-2 as crss = 0 + A D
= (0.069 MPa) +
( 6.21
x 10-3 MPa - mm ) 106 mm -2 = 6.28 MPa (910 psi)
Recovery Recrystallization Grain Growth
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8.31 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 strain-free grains forms, and the material becomes softer and more ductile.
8.32 We are asked to estimate the fraction of recrystallization from the photomicrograph in Figure 8.21c. 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 672 total squares, the specimen is about 60% recrystallized.
8.33
During cold-working, 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.
8.34 (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.
8.35 In this problem, we are asked for the length of time required for the average grain size of a brass material to increase a specified amount using Figure 8.25.
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(a) At 500C, the time necessary for the average grain diameter to increase from 0.01 to 0.1 mm is approximately 3500 min. (b) At 600C the time required for this same grain size increase is approximately 150 min.
8.36 (a) Using the data given and Equation 8.8 (taking n = 2), we may set up two simultaneous equations with d0 and K as unknowns; thus
2 (3.9 x 10-2 mm) 2 - d 0 = (30 min) K 2 x (6.6 10-2 mm) 2 - d 0 = (90 min) K
Solution of these expressions yields a value for d0, the original grain diameter, of d0 = 0.01 mm, and a value for K of 4.73 x 10-5 mm2/min (b) At 150 min, the diameter d is computed using a rearranged form of Equation 8.8 as
d =
2 d 0 + Kt
=
(0.01 mm) 2 + ( 4.73 x 10-5 mm 2 / min ) (150 min) = 0.085 mm
8.37 Yes, it is possible to reduce the average grain diameter of an undeformed alloy specimen from 0.050 mm to 0.020 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 0.020 mm.
8.38 (a) The temperature dependence of grain growth is incorporated into the constant K in Equation 8.8. (b) The explicit expression for this temperature dependence is of the form Q K = K 0 exp - RT
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in which K0 is a temperature-independent constant, the parameter Q is an activation energy, and R and T are the gas constant and absolute temperature, respectively.
8.39 This problem calls for us to calculate the yield strength of a brass specimen after it has been heated to an elevated temperature at which grain growth was allowed to occur; the yield strength (150 MPa) was given at a grain size of 0.01 mm. It is first necessary to calculate the constant ky in Equation 8.6 as
ky =
y - 0 d -1/2 = 12.5 MPa - mm 1 / 2
=
150 MPa - 25 MPa (0.01 mm)-1 / 2
Next, we must determine the average grain size after the heat treatment. From Figure 8.25 at 500C after 1000 s (16.7 min) the average grain size of a brass material is about 0.016 mm. Therefore, calculating y at this new grain size using Equation 8.6 we get y = 0 + k yd
-1/2
= 25 MPa + (12.5 MPa - mm 1/2 ) (0.016 mm) -1/2 = 124 MPa (18, 000 psi)
Crystalline Ceramics (Deformation Mechanisms for Ceramic Materials)
8.40 Crystalline ceramics are harder yet more brittle than metals because they (ceramics) have fewer slip systems, and, therefore, dislocation motion is highly restricted.
Deformation of Semicrystalline Polymers (Deformation of Elastomers)
8.41 (a) and (b) The mechanisms by which semicrystalline polymers elastically and plastically deform are described in Section 8.17. (c) The explanation of the mechanism by which elastomers elastically deform is provided in Section 8.19.
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Factors That Influence the Mechanical Properties of Semicrystalline Polymers Deformation of Elastomers
8.42 (a) The tensile modulus is not directly influenced by a polymer's molecular weight. (b) Tensile modulus increases with increasing degree of crystallinity for semicrystalline polymers. This is due to enhanced secondary interchain bonding which results from adjacent aligned chain segments as percent crystallinity increases. This enhanced interchain bonding inhibits relative interchain motion. (c) Deformation by drawing also increases the tensile modulus. The reason for this is that drawing produces a highly oriented molecular structure, and a relatively high degree of interchain secondary bonding. (d) When an undeformed semicrystalline polymer is annealed below its melting temperature, the tensile modulus increases. (e) A drawn semicrystalline polymer that is annealed experiences a decrease in tensile modulus as a result of a reduction in chain-induced crystallinity, and a reduction in interchain bonding forces.
8.43 (a) The tensile strength of a semicrystalline polymer increases with increasing molecular weight. This effect is explained by increased chain entanglements at higher molecular weights. (b) Increasing the degree of crystallinity of a semicrystalline polymer leads to an enhancement of the tensile strength. Again, this is due to enhanced interchain bonding and forces; in response to applied stresses, interchain motions are thus inhibited. (c) Deformation by drawing increases the tensile strength of a semicrystalline polymer. This effect is due to the highly oriented chain structure that is produced by drawing, which gives rise to higher interchain secondary bonding forces. (d) Annealing an undeformed semicrystalline polymer produces an increase in its tensile strength.
8.44 Normal butane has a higher melting temperature as a result of its molecular structure (Section 4.2). There is more of an opportunity for van der Waals bonds to form between two molecules in close proximity to one another than for isobutane because of the linear nature of each normal butane molecule.
8.45 This problem gives us the tensile strengths and associated number-average molecular weights for two poly(methyl methacrylate) materials and then asks that we estimate the tensile strength for M n = 30,000 g/mol. Equation 8.10 cites the dependence of the tensile strength on M n . Thus, using the data provided in the problem,
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we may set up two simultaneous equations from which it is possible to solve for the two constants TS and A. These equations are as follows: A 40, 000 g / mol A 60, 000 g / mol
107 MPa = TS -
170 MPa = TS -
Thus, the values of the two constants are: TS = 296 MPa and A = 7.56 x 106 MPa-g/mol. Substituting these values into Equation 8.10 for M n = 30,000 g/mol leads to TS = TS - A 30, 000 g / mol
= 296 MPa -
7.56 x 106 MPa - g / mol 30, 000 g / mol
= 44 MPa
8.46 This problem gives us the tensile strengths and associated number-average molecular weights for two polyethylene materials and then asks that we estimate the M n that is required for a tensile strength of 195 MPa. Equation 8.10 cites the dependence of the tensile strength on M n . Thus, using the data provided in the problem, we may set up two simultaneous equations from which it is possible to solve for the two constants TS and A. These equations are as follows: A 12, 700 g / mol A 28, 500 g / mol
85 MPa = TS -
150 MPa = TS -
Thus, the values of the two constants are: TS = 202 MPa and A = 1.489 x 106 MPa-g/mol. Solving for M n in Equation 8.10 and substituting TS = 195 MPa as well as the above values for TS and A leads to A TS - TS
Mn =
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=
1.489 x 106 MPa - g / mol = 213, 000 g/mol 202 MPa - 195 MPa
8.47 For each of four pairs of polymers, we are to do the following: (1) determine whether or not it is possible to decide which has the higher tensile modulus; (2) if so, note which has the higher tensile modulus and then state the reasons for this choice; and (3) if it is not possible to decide, then state why. (a) Yes, it is possible. The linear and isotactic poly(vinyl chloride) will display a greater tensile modulus. Linear polymers are more likely to crystallize that branched ones. In addition, polymers having isotactic structures will normally have a higher degree of crystallinity that those having atactic structures. Increasing a polymer's crystallinity leads to an increase in its tensile modulus. In addition, tensile modulus is independent of molecular weight--the atactic/branched material has the higher molecular weight. (b) Yes, it is possible. The block styrene-butadiene copolymer with 10% of possible sites crosslinked will have the higher modulus. Block copolymers normally have higher degrees of crystallinity than random copolymers of the same material. A higher degree of crystallinity favors larger moduli. In addition, the block copolymer also has a higher degree of crosslinking; increasing the amount of crosslinking also enhances the tensile modulus. (c) No, it is not possible. Branched polyethylene will tend to have a low degree of crystallinity since branched polymers don't normally crystallize. The atactic polypropylene probably also has a relatively low degree of crystallinity; atactic structures also don't tend to crystallize, and polypropylene has a more complex mer structure than does polyethylene. Tensile modulus increases with degree of crystallinity, and it is not possible to determine which polymer is more crystalline. Furthermore, tensile modulus is independent of molecular weight.
8.48 For each of four pairs of polymers, we are to do the following: (1) determine whether or not it is possible to decide which has the higher tensile strength; (2) if it is possible, then note which has the higher tensile strength and then state the reasons for this choice; and (3) if it is not possible to decide, to state why. (a) Yes, it is possible. The linear and isotactic material will have the higher tensile strength. Both linearity and isotacticity favor a higher degree of crystallinity than do branching and atacticity; and tensile strength increases with increasing degree of crystallinity. Furthermore, the molecular weight of the linear/isotactic material is higher (100,000 g/mol versus 75,000 g/mol), and tensile strength increases with increasing molecular weight. (b) No, it is not possible. Alternating copolymers tend to be more crystalline than graft copolymers, and tensile strength increases with degree of crystallinity. However, the graft material has a higher degree of crosslinking, and tensile strength increases with the percentage of crosslinks. (c) Yes, it is possible. The network polyester will display a greater tensile strength. Relative chain motion is much more restricted than for the lightly branched polytetrafluoroethylene since there are many more of the strong covalent bonds for the network structure.
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8.49 The strength of a polychlorotrifluoroethylene having the mer structure
will be greater than for a polytetrafluoroethylene having the same molecular weight and degree of crystallinity. The replacement of one fluorine atom within the PTFE mer with a chlorine atom leads to a higher interchain attraction, and, thus, a stronger polymer. Furthermore, poly(vinyl chloride) is stronger than polyethylene (Table 7.2) for the same reason.
8.50 (a) Shown below are the stress-strain curves for the two polyisoprene materials, both of which have a molecular weight of 100,000 g/mol. These two materials are elastomers and will have curves similar to curve C in Figure 7.22. However, the curve for the material having the greater number of crosslinks (20%) will have a higher elastic modulus at all strains.
(b) Shown below are the stress-strain curves for the two polypropylene materials. These materials will most probably display the stress-strain behavior of a normal plastic, curve B in Figure 7.22. However, the syndiotactic polypropylene has a higher molecular weight and will also undoubtedly have a higher degree of crystallinity; therefore, it will have a higher strength.
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(c) Shown below are the stress-strain curves for the two polyethylene materials. crosslinked polyethylene will be stiffer, stronger, and more brittle (curve A of Figure 7.22).
The branched
polyethylene will display the behavior of a normal plastic, curve B in Figure 7.22. On the other hand, the heavily
8.51 Two molecular characteristics essential for elastomers are: (1) they must be amorphous, having chains that are extensively coiled and kinked in the unstressed state; and (2) there must be some crosslinking.
8.52 This question asks us to choose from a list of materials those which would be expected to be elastomers and those which would be thermosetting polymers. (a) Linear and crystalline polyethylene would be neither an elastomer nor a thermoset since it is a linear polymer.
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(b) Phenol-formaldehyde having a network structure would be a thermosetting polymer since it has a network structure. It would not be an elastomer since it does not have a crosslinked chain structure. (c) Heavily crosslinked polyisoprene having a glass transition temperature of 50C would be a thermosetting polymer because it is heavily crosslinked. It would not be an elastomer since it is heavily crosslinked and room temperature is below its Tg. (d) Lightly crosslinked polyisoprene having a glass transition temperature of 60C is both an elastomer and a thermoset. It is an elastomer because it is lightly crosslinked and has a Tg below room temperature. It is a thermoset because it is crosslinked. (e) Linear and partially amorphous poly(vinyl chloride) is neither an elastomer nor a thermoset. In order to be either it must have some crosslinking.
8.53 This problem asks that we compute the fraction of possible crosslink sites in 10 kg of polybutadiene when 4.8 kg of S is added, assuming that, on the average, 4.5 sulfur atoms participate in each crosslink bond. Given the butadiene mer unit in Table 4.5, we may calculate its molecular weight as follows: A(butadiene) = 4(AC) + 6(AH)
= (4)(12.01 g/mol) + 6(1.008 g/mol) = 54.09 g/mol 10, 000 g = 184.9 mol. 54.09 g / mol
Which means that in 10 kg of butadiene there are
For the vulcanization of polybutadiene, there are two possible crosslink sites per mer--one for each of the two carbon atoms that are doubly bonded. Furthermore, each of these crosslinks forms a bridge between two mers. Therefore, we can say that there is the equivalent of one crosslink per mer. Therefore, let us now calculate the number of moles of sulfur (nsulfur) that react with the butadiene, by taking the mole ratio of sulfur to butadiene, and then dividing this ratio by 4.5 atoms per crosslink; this yields the fraction of possible sites that are crosslinked. Thus 4800 g = 149.7 mol 32.06 g / mol
nsulfur = And
149.7 mol fraction sites crosslinked = 184.9 mol = 0.180 4.5
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8.54 For an alternating chloroprene-acrylonitrile copolymer, we are asked to compute the weight percent sulfur necessary for complete crosslinking, assuming that, on the average, five sulfur atoms participate in each crosslink. The chloroprene and acrylonitrile mers are shown in Table 4.5, from which it may be noted that there are two possible crosslink sites on each chloroprene mer (one site at each of the two carbon atoms that are doubly bonded), and no possible sites for acrylonitrile; also, since it is an alternating copolymer, the ratio of chloroprene to acrylonitrile mers is 1:1. Thus, for each pair of combined chloroprene-acrylonitrile mers which crosslink, ten sulfur atoms are required, or, for complete crosslinking, the sulfur-to-(chloroprene-acrylonitrile) ratio is 5:1. Now, let us consider as our basis, one mole of the combined chloroprene-acrylonitrile mers. In order for complete crosslinking, five moles of sulfur are required. Thus, for us to convert this composition to weight percent, it is necessary to convert moles to mass. The acrylonitrile mer consists of three carbon atoms, three hydrogen atoms, and one nitrogen atom; the chloroprene mer is composed of four carbons, five hydrogens, and one chlorine. This gives a molecular weight for the combined mer of
m(chloroprene-acrylonitrile) = 3(AC) + 3(AH) + AN + 4(AC) + 5(AH) + ACl
= 7(12.01 g/mol) + 8(1.008 g/mol) + 14.007 g/mol + 35.45 g/mol = 141.59 g/mol Or, in one mole of this combined mer, there are 141.59 g. Furthermore, for complete crosslinking 5.0 mol of sulfur is required, which amounts to (5.0 mol)(32.06 g/mol) = 160.3 g. Thus, the concentration of S in weight percent CS is just 160.3 g x 100 = 53.1 wt% 160.3 g + 141.59 g
CS =
8.55 This problem asks for us to determine how many crosslinks form per isoprene mer when 57 wt% sulfur is added. If we arbitrarily consider 100 g of the vulcanized material, 57 g will be sulfur and 43 g will be polyisoprene. Next, let us find how many moles of sulfur and isoprene correspond to these masses. The atomic weight of sulfur is 32.06 g/mol, and thus, 57 g = 1.78 mol 32.06 g / mol
# moles S =
However, there are 6 sulfur atoms in each crosslink, which requires us to divide the number of moles of sulfur by 6 in order to get the number of moles of sulfur per crosslink, which is equal to 0.297 moles. Now, in each isoprene mer unit there are five carbon atoms and eight hydrogen atoms. Thus, the molecular weight of a mole of isoprene units is
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(5)(12.01 g/mol) + (8)(1.008 g/mol) = 68.11 g/mol Or, in 43 g of polyisoprene 43 g = 0.631 mol 68.11 g / mol
# moles isoprene =
Therefore, the ratio of moles of S per crosslink to the number of moles of polyisoprene is 1.78 mol : 1 = 2.82 : 1 0.631 mol When all possible sites are crosslinked, the ratio of the number of moles of sulfur to the number of moles of isoprene is 6:1, since there are two crosslink sites per mer unit and each crosslink is shared between mers on adjacent chains, and there are 6 sulfur atoms per crosslink. Finally, to determine the fraction of sites that are crosslinked, we just divide the actual crosslinked sulfur/isoprene ratio by the completely crosslinked ratio. Or, 2.82 / 1 = 0.47 6/1
fraction of mer sites crosslinked =
8.56 We are asked what weight percent of sulfur must be added to polyisoprene in order to ensure that 8% of possible sites are crosslinked, assuming that, on the average, three sulfur atoms are associated with each crosslink. Table 4.5 shows the chemical repeat unit for cis-isoprene. For each of these units there are two possible crosslink sites; one site is associated with each of the two carbon atoms that are involved in the chain double bond. Since 8% of the possible sites are crosslinked, for each 100 isoprene mers 8 of them are crosslinked; actually there are two crosslink sites per mer, but each crosslink is shared by two chains. Furthermore, on the average we assume that each crosslink is composed of 3 sulfur atoms; thus, there must be 3 x 8 or 24 sulfur atoms added for every 100 isoprene mers. In terms of moles, it is necessary to add 24 moles of sulfur to 100 moles of isoprene. The atomic weight of sulfur is 32.06 g/mol, while the molecular weight of isoprene is 5(AC) + 8(AH)
= (5)(12.01 g/mol) + (8)(1.008 g/mol) = 68.11 g/mol The mass of sulfur added (mS) is
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mS = (24 mol)(32.06 g/mol) = 769.4 g
While for isoprene
mip = (100 mol)(68.11 g/mol) = 6811 g
Or, the concentration of sulfur in weight percent is just 769.4 g x 100 = 10.2 wt% 769.4 g + 6811 g
CS =
8.57 The reaction by which a chloroprene rubber may become vulcanized is as follows:
DESIGN PROBLEMS
Strain Hardening Recrystallization
8.D1 This problem calls for us to determine whether or not it is possible to cold work steel so as to give a minimum Brinell hardness of 240 and a ductility of at least 15%EL. According to Figure 7.31, a Brinell hardness of 240 corresponds to a tensile strength of 800 MPa (116,000 psi). Furthermore, from Figure 8.19(b), in order to achieve a tensile strength of 800 MPa, deformation of at least 13%CW is necessary. Finally, if we cold work the steel to 13%CW, then the ductility is 15%EL from Figure 8.19(c). Therefore, it is possible to meet both of these criteria by plastically deforming the steel.
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8.D2 We are asked to determine whether or not 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. According to Figure 7.31, a Brinell hardness of 150 corresponds to a tensile strength of 500 MPa (72,000 psi.) Furthermore, from Figure 8.19(b), in order to achieve a tensile strength of 500 MPa, deformation of at least 36%CW is necessary. Finally, if we are to achieve a ductility of at least 20%EL, then a maximum deformation of 23%CW is possible from Figure 8.19(c). Therefore, it is not possible to meet both of these criteria by plastically deforming brass.
8.D3 (a) For this portion of the problem we are to determine the ductility of cold-worked steel that has a Brinell hardness of 240. From Figure 7.31, a Brinell hardness of 240 corresponds to a tensile strength of 820 MPa (120,000 psi), which, from Figure 8.19(b), requires a deformation of 17%CW. Furthermore, 17%CW yields a ductility of about 13%EL for steel, Figure 8.19(c). (b) We are now asked to determine the radius after deformation if the uncold-worked radius is 10 mm (0.40 in.). From Equation 8.7 and for a cylindrical specimen
2 2 r0 - r d %CW = r2 0
Now, solving for rd from this expression, we get
x 100
%CW rd = r0 1 - 100 17 = 9.11 mm (0.364 in.) 100
= (10 mm) 1 -
8.D4 This problem asks us to determine which of copper, brass, and a 1040 steel may be cold-worked so as to achieve a minimum yield strength of 345 MPa (50,000 psi) while maintaining a minimum ductility of 20%EL. For each of these alloys, the minimum cold work necessary to achieve the yield strength may be determined from Figure 8.19(a), while the maximum possible cold work for the ductility is found in Figure 8.19(c). These data are tabulated below. Yield Strength (> 345 MPa) Steel Brass Any %CW > 20%CW Ductility (> 20%EL) < 5%CW < 23%CW
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Copper
> 54%CW
< 15%CW
Thus, both the 1040 steel and brass are possible candidates since for these alloys there is an overlap of percents coldwork to achieve the required minimum yield strength and ductility values.
8.D5 This problem calls for us to explain the procedure by which a cylindrical rod of steel may be deformed so as to produce a given final diameter, as well as a specific tensile strength and ductility. First let us calculate the percent cold work and attendant tensile strength and ductility if the drawing is carried out without interruption. From Equation 8.7
%CW =
d 2 d 0 - d 2 2 d 2 0 2
2
x 100
=
15.2 mm 2 10 mm 2 - 2 2 15.2 mm 2 2
x 100 = 56%CW
At 56%CW, the steel will have a tensile strength on the order of 920 MPa (133,000 psi) [Figure 8.19(b)], which is adequate; however, the ductility will be less than 10%EL [Figure 8.19(c)], which is insufficient. Instead of performing the drawing in a single operation, let us initially draw some fraction of the total deformation, then anneal to recrystallize, and, finally, cold-work the material a second time in order to achieve the final diameter, tensile strength, and ductility. Reference to Figure 8.19(b) indicates that 20%CW is necessary to yield a tensile strength of 840 MPa (122,000 psi). Similarly, a maximum of 21%CW is possible for 12%EL [Figure 8.19(c)]. The average of these
' extremes is 20.5%CW. If the final diameter after the first drawing is d 0 , then
20.5%CW =
' 2 10 mm d 0 - 2 2 ' d 0 2
2
2
x 100
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' ' And, solving for d 0 , yields d 0 = 11.2 mm (0.45 in.).
8.D6 Let us first calculate the percent cold work and attendant yield strength and ductility if the drawing is carried out without interruption. From Equation 8.7 d 2 d 2 0 - d 2 2 d 2 0 2
%CW =
x 100
=
10.2 mm 2 7.6 mm 2 - 2 2 10.2 mm 2 2
x 100 = 44.5%CW
At 44.5%CW, the brass will have a yield strength on the order of 420 MPa (61,000 psi), Figure 8.19(a), which is adequate; however, the ductility will be about 5%EL, Figure 8.19(c), which is insufficient. Instead of performing the drawing in a single operation, let us initially draw some fraction of the total deformation, then anneal to recrystallize, and, finally, cold work the material a second time in order to achieve the final diameter, yield strength, and ductility. Reference to Figure 8.19(a) indicates that 26%CW is necessary to give a yield strength of 380 MPa. Similarly, a maximum of 27.5%CW is possible for 15%EL [Figure 8.19(c)]. The average of these two values is
' 26.8%CW, which we will use in the calculations. If the final diameter after the first drawing is d 0 , then
26.8%CW =
' 2 7.6 mm d 0 - 2 2 ' d 0 2
2
2
x 100
' ' And, solving for d 0 yields d 0 = 9.4 mm (0.37 in.).
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8.D7 This problem calls for us to cold work some brass stock that has been previously cold worked in order to achieve minimum tensile strength and ductility values of 450 MPa (65,000 psi) and 13%EL, respectively, while the final diameter must be 12.7 mm (0.50 in.). Furthermore, the material may not be deformed beyond 65%CW. Let us start by deciding what percent coldwork is necessary for the minimum tensile strength and ductility values, assuming that a recrystallization heat treatment is possible. From Figure 8.19(b), at least 27%CW is required for a tensile strength of 450 MPa. Furthermore, according to Figure 8.19(c), 13%EL corresponds a maximum of 30%CW. Let us take the average of these two values (i.e., 28.5%CW), and determine what previous specimen diameter is required to yield a final diameter of 12.7 mm. For cylindrical specimens, Equation 8.7 takes the form d 2 d 2 0 - d 2 2 d 2 0 2
%CW =
x 100
Solving for the original diameter d0 yields dd % CW 1- 100 12.7 mm 1 - 0.285
d0 =
=
= 15.0 mm (0.591 in.)
Now, let us determine its undeformed diameter realizing that a diameter of 19.0 mm corresponds to 35%CW. Again solving for d0 using the above equation and assuming dd = 19.0 mm yields dd % CW 1- 100 19.0 mm 1 - 0.35
d0 =
=
= 23.6 mm (0.930 in.)
At this point let us see if it is possible to deform the material from 23.6 mm to 15.0 mm without exceeding the 65%CW limit. Again employing Equation 8.7 23.6 mm 2 15.0 mm 2 - 2 2 23.6 mm 2 2
%CW =
x 100 = 59.6%CW
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In summary, the procedure which can be used to produce the desired material would be as follows: cold work the as-received stock to 15.0 mm (0.591 in.), heat treat it to achieve complete recrystallization, and then cold work the material again to 12.7 mm (0.50 in.), which will give the desired tensile strength and ductility.
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