E 45 - Fall 2010 - Gronsky - Midterm 1 (solution)

M 01 233 433 533 3632 3 3634 e45 fall 10 midterm 01

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Without the ASTM standard, these terms would have nebulous meaning. !"#$%& ( ¡ )* +,-+, !"#.//( m )*01$ 233 433 533 3632 3 3634 E45 Fall 10 Midterm 01 Solutions Professor R. Gronsky page2 of 11 σ e ε e σ t ε t
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1. Mechanical Properties c . What value would you specify as the elastic modulus of this alloy? Explain . The "elastic" modulus, also known as "Young's Modulus" ( E ) is the slope of the initial portion of the stress-strain curve where Hooke's Law applies. Both σ e ε e and σ t ε t curves share this linear behavior, which is due to the stretching of atomic bonds, before any bond-breaking commences. It can be calculated directly from the plot, using convenient values such as the following d . What value would you specify for the ultimate tensile strength of the alloy? Explain . This is read directly from the σ e ε e curve. It is the "peak" value at the inflection point, shown on the figure above to be UTS = 200 MPa . !"#$%& ( ¡ )* +,-+, !"#.//( m )*01$ 233 433 533 3632 3 3634 σ = E ε E = σ ε = [100 0] MPa [0 . 002 0] cm / cm = 50 GPa E45 Fall 10 Midterm 01 Solutions Professor R. Gronsky page3 of 11
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2. Bonding a . Simeon-Denis Poisson was a French mathematician who related elastic strain normal to the stress axis ( ε x ) to elastic strain along the stress axis ( ε z ) as a ratio, which now bears his name (Poisson's ratio), and is written as Type 304 stainless steel is a high strength metallic alloy with an fcc structure and υ = 0.29, while aluminum bronze is a lower strength metallic alloy with an fcc structure and υ = 0.33. Explain , citing concepts of metallic bonding. Metallic bonds result from the delocalization of the valence electrons that participate in bonding, enabling global "sharing" of those electrons among many ion cores, generating bonds that are "non-directional." The observed differences in Poisson's ratio between two "metallically bonded" solids (steel and bronze both exhibit the properties expected of "metals") suggests that this is not always the case, and some directionality may be at play. Under these circumstances, the accepted explanation is an intrusion of some "covalent character" to the bonds, as also used to rationalize the CN=8 in BCC metals, for example. b . Alumina (Al 2 O 3 ) is an oxide ceramic formed from trivalent aluminum cations and divalent oxygen anions. The radius ratio of cation to anion ( r/R ) in alumina is 0.4318, in the range 0.414 r/R 0.732, associated with a coordination number of 6. Explain how the nature of ionic bonding causes the coordination number to be a prominent consideration in the crystal structure of oxide ceramics. Ionic bonding is the result of "charge transfer," which creates two oppositely charged ions that succumb to the influence of Coulombic forces to establish a "bond" at equilibrium (zero force condition). Such a bond is "non-directional" suggesting that bonding can occur in any direction. However, during charge transfer, cations decrease in size, and anions increase in size, imposing a size effect on how they can pack together in the solid state. This packing geometry is limited by the radius ratio, and the resulting coordination number, influencing the development of crystal structure in ionic solids.
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