Working With Ashby Plots: Example Maximize M f

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Working With Ashby Plots Example Maximize M
?f/?
For fixed M log M log ?f log ?
constant C

• With strength ?f (MPa), and ? (Mg/m3), what
  are units of ?f/??
• For given M, use log ? f vs. log ?.
• Hence, with log ?f log ? C, need lines of
  slope 1.
• All lines of slope 1 have the different M
  values!
• All materials on a give line have the same M but
  NOT the same materials properties (?f or ? ),
  e.g. some less dense (lighter).

From Materials Selection for Mechanical
Design Ed. 2, M. Ashby
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Ashby Plots Strength (?f ) versus Density (?)
Strength can be YS for metals and polymer,
compressive strength for ceramics, tear strength
for elastomers, and TS for composites, for
example.
Taken from Materials Selection for Mechanical
Design Ed. 2, M. Ashby
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Where are lines of slope 1 with specified M?

• With Materials Index M ?f/?.
• Give some values of M 10 Pa/g/m3.
• Connect those values by a line. What is slope of
  this log(?f) - log(?) line?
• Note slope is ?log(?)/?log(?).
• Give some values of M100 Pa/g/m3.
• Why does the M 100 line have 10 times less
  mass than M 10 line?
• Because mass ? 1/M ? ?.

From Materials Selection for Mechanical
Design Ed. 2, M. Ashby
4
Material for Torsionally Stressed Shaft

• As derived in book, minimizing mass (m) when
  considering shear strength (?f) one finds.

Strength (MPa)

• Get min. mass by maximizing materials
  performance index M

• Look at lines with slope 3/2
• log ?f (3/2) log ? (3/2)C

Density (Mg/m3)
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Ashby Plot Strength vs Density

• What are the M values of the four lines mark?
• - Hint use values of ?f and ? that are labeled
  already.
• What are the slopes of those four lines?
• Hint count decades of rise and runs, the ratio
  is slope of ?log?f /?log ? (3/2).

Strength (MPa)

• All materials along a line have same
  performance.
• But, M 30 has 1/3 the mass of M 10, because
  the mass ? 1/M ? ?.

Density (Mg/m3)
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Calculating Performance Values and Slopes

• Lowest Line
• Log-log plot is harder to read, obtain M value
  with values at tick marks, e.g. ? 0.1 Mg/m3 and
  ?f 0.167 MPa or ? 0.3 Mg/m3 and ?f 0.867
  MPa.
• Performance Value M (?f)2/3/? (0.867
  MPa)2/3/0.3 Mg/m3
• M 3 (MPa)2/3/Mg/m3.
• Slope of 3/2 move right 2 decades on ? and up 3
  decades on ?f.
• You can verify using (0.867 MPa, 0.3 Mg/m3) and
  (867 MPa, 30 Mg/m3).

• Next Line
• ?0.1 Mg/m3 and ?f1 MPa, M (1MPa)2/3/0.1
  Mg/m3 10 (MPa)2/3/Mg/m3.
• Again slope of 3/2 is 2 decades on ? and up 3
  decades on ?f.