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Unformatted text preview: includes x, a and c/2 as its sides permits the side lengths to be summed as a2 = x2 + (c/2)2 Substituting,
(c/2) = a −
2 2 and solving, a/2 cos 30◦ 2 c/a = 1.633 Now note that the ratio of the Burgers vectors along the two directions of interest is exactly equal to the c/a ratio.
b[0001] = c a b[11¯ 20] Consequently the relative dislocation energies differ by a smaller factor,
b2 [0001] = (1.633)2 = 2.67 b2 [11¯ 20] Problem 4.23 Solution Professor R. Gronsky page 3 of 3 Problem 4.25
J. F. Shackelford, Introduction to Materials Science for Engineers, 7th Edition, Prentice Hall, New Jersey (2009) 4.25 Determine the grainsize number, G, for the microstructure shown in Figure 418. (Keep in mind that the precise answer will depend on your choice of an area of sampling.) nλ = 2dhkl sin θ
SOLUTION a dhkl = √ 2 + k 2 + l2 h Using the procedure outlined in Figure 420 (p. 116 of the text) for the calculation of the ASTM grain size number (G), begin by drawing a circle of known dimensions in the center of micrograph shown in Figure 418, which was recorded at the required magnification of 100× (NOTE: nλ 2 2 2 sin θ = a micrograph recorded at a different magnificationh + k require a magnification correction). would + l 2a The enclosing circle shown here has a diameter of 2.25 inches. ρatoms = 2.33 Mg 1 mol 106 g 0.6023 × 102 4 atoms × × × = 5.00×1028 atoms/m3 3 m 28.09 g Mg mol ρvacancies = 1 × 107 vacancies/atom × 5.00 × 1028 atoms/m3 b111  = b110  = √ 3a 2 2a √ b001  = a b110 2 2 circle and 39 cut by the circumference, which are 8 There are 77 grains completely enclosed by thea2 = 3 a2 = = 2.67 2 of 96.5 grains. The area density (N) of grains is next counted as 39/2 or 19.5 grains, forba total  111  3 4 calculated as follows: N= 96.5 grains grains = 24.3 2 in2 π (2.25/2) in2 Problem 4.25 Solution Professor R. Gronsky page 1 of 2 b001  = a
Problem 4.25 4 b110 Materials Science for Engineers, 7th Edition, Prentice Hall, New Jersey (2009) 2a 8 J. F. Shackelford, Intr...
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This note was uploaded on 01/21/2011 for the course E 45 taught by Professor Gronsky during the Fall '08 term at University of California, Berkeley.
 Fall '08
 GRONSKY

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