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ts2201f05

ts2201f05 - EMSE 201 Introduction to Ma teria ls Science...

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EMSE 201 — Introduction to Materials Science & Engineering 9 November 2005 Name: S O L U T I O N Department of Materials Science and Engineering 1 of 6 Case Western Reserve University Test #2 — 75 minutes; 150 points; 6 questions; 6 pages; 15% of course grade No calculators or formula sheets are allowed. Where numerical answers are requested, full credit will be given for correctly setting up the calculation and for specifying the correct units . Use the backs of these sheets if necessary. 1) (28 points) For the brass alloy whose engineering stress-strain curve is shown at right, indicate how one would determine the following properties, and set up the calculation (with units ): a) modulus of elasticity b) yield strength at a strain offset of 0.2% c) ultimate tensile strength d) strain to failure e) modulus of resilience f) modulus of toughness (This is Callister, example 6.3) a) modulus of elasticity (E) is the slope of the linear region of the curve. From the inset, a strain of 0.002 corresponds to a stress of 25 ksi or 175 MPa, giving a modulus of: E = (25±2)/.002 ksi (= 12 × 10 6 psi) or (175±15)/.002 MPa (= 87 GPa). ( 1 pt stress + 1 pt strain + 2 pts indication/set up + 1 pt units). b) As indicated in the inset, draw a line parallel to the linear region of the curve that passes through 0.2% on the strain axis ( 2 pts ). The stress at which this line intersects the curve ( 1 pt ) is the offset yield strength, σ y : σ y = 250±20 MPa or 36±4 ksi here (1 pt + 1 pt units).

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EMSE 201 — Introduction to Materials Science & Engineering 9 November 2005 Department of Materials Science and Engineering 2 of 7 Case Western Reserve University c) Ultimate tensile strength, σ ut , is the stress at the highest point on the stress-strain curve ( 1 pt ): 450±20 MPa or 65±3 ksi (1 pt + 1 pt units). d) Strain to failure, ε f , is the strain at the end of the curve (point of fracture) ( 1 pt ): ε f = 37±1% here (1 pt + 1 pt units). e) Modulus of resilience, the area under the elastic (linear) portion of the stress-strain curve ( 2 pts partial credit ), can be approximated as σ y 2 /2E ( 4 pts ). Plugging in numbers ( 1 pt ) gives: 0.36 MPa or 54 psi (1 pt units) Alternatively, the area under the linear portion of the curve can be taken from the inset as σ pl ε pl /2 ( 4 pts ). This gives: 210 MPa × 0.0025/2 = 0.26 MPa or 30 ksi × 0.0025/2 = 38 psi (1 pt + 1 pt units). These values are lower because they don’t overestimate the proportional limit by using the yield strength. f) Modulus of toughness is the area under the total engineering stress-strain curve ( 2 pts partial credit ), estimated as σ ut ε f or more closely as ( σ ut + σ y ) ε f /2 ( 4 pts for either). Plugging in numbers ( 1 pt + 1 pt units ) gives: σ ut ε f ( σ ut + σ y ) ε f /2 450 MPa × 0.37 = 165 MPa or (450 + 250) MPa × 0.37/2 = 130 MPa 65 ksi × 0.37 = 24 ksi (65 + 36) ksi × 0.37 = 19 ksi Note that the units on both of these last two moduli, though equivalent to units of stress, are better thought of as energy per unit volume.
EMSE 201 — Introduction to Materials Science & Engineering 9 November 2005 Department of Materials Science and Engineering 3 of 7 Case Western Reserve University 2)

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