## Appendix Catalogue of material indices

Material indices help to identify the applications in which a material might excel and those for generic components are assembled here. Their derivation and use is illustrated in Chapters 5, 11 and 12.

(a) Stiffness-limited design at minimum mass (cost, energya)

Function and constraintsa Maximizeb

SHAFT (loaded in torsion)

Stiffness, length, shape specified, section area free G1/2/p

Stiffness, length, outer radius specified; wall thickness free G/p

Stiffness, length, wall thickness specified, outer radius free G1/3/p

BEAM (loaded in bending)

Stiffness, length, shape specified; section area free E1/2/p

Stiffness, length, height specified; width free E/p

Stiffness, length, width specified; height free E1/3/p

COLUMN (compression strut, failure by elastic buckling) E1/2/p buckling load, length, shape specified; section area free aTo minimize cost, use the above criteria for minimum weight, replacing density p by Cmp, where Cm is the material cost per kg. To minimize energy content, use the above criteria for minimum weight replacing density p by qp where q is the energy content per kg.

bE = Young's modulus; G = shear modulus; p = density.

(b) Stiffness-limited design at minimum mass (cost, energya)

Function and constraintsa Maximizeb

PANEL (flat plate, loaded in bending) E1/3/p stiffness, length, width specified, thickness free

PLATE (flat plate, compressed in-plane, buckling failure) E1/3/p collapse load, length and width specified, thickness free

Function and constraintsa

Maximizeb

CYLINDER WITH INTERNAL PRESSURE

elastic distortion, pressure and radius specified; wall E/p thickness free

### SPHERICAL SHELL WITH INTERNAL PRESSURE

elastic distortion, pressure and radius specified, wall E/(1 — v)p thickness free aTo minimize cost, use the above criteria for minimum weight, replacing density p by Cmp, where Cm is the material cost per kg. To minimize energy content, use the above criteria for minimum weight replacing density p by qp where q is the energy content per kg.

bE = Young's modulus; G = shear modulus; p = density.

(c) Strength-limited design at minimum mass (cost, energya)

Function and constraints^ Maximizeb

SHAFT (loaded in torsion)

Load, length, shape specified, section area free of/3/p

Load, length, outer radius specified; wall thickness free of/p

Load, length, wall thickness specified, outer radius free of /p

BEAM (loaded in bending)

Load, length, shape specified; section area free of/3/p

Load, length, height specified; width free of/p

Load, length, width specified; height free of /p

COLUMN (compression strut)

Load, length, shape specified; section area free of/p

PANEL (flat plate, loaded in bending)

Stiffness, length, width specified, thickness free of /p aTo minimize cost, use the above criteria for minimum weight, replacing density p by Cmp, where Cm is the material cost per kg. To minimize energy content, use the above criteria for minimum weight replacing density p by qp where q is the energy content per kg.

bOf = failure strength (the yield strength for metals and ductile polymers, the tensile strength for ceramics, glasses and brittle polymers); p = density.

cFor design for infinite fatigue life, replace of by the endurance limit oe.

(d) Strength-limited design at minimum mass (cost, energya)

Function and constraints^ |
Maximizeb |

PLATE (flat plate, compressed in-plane, buckling failure) |
1/2 / f /p |

Collapse load, and width specified, thickness free | |

CYLINDER WITH INTERNAL PRESSURE | |

Elastic distortion, pressure and radius specified; wall |
Of/p |

thickness free | |

SPHERICAL SHELL WITH INTERNAL PRESSURE | |

Elastic distortion, pressure and radius specified, wall |
Of/p |

thickness free | |

FLYWHEELS, ROTATING DISKS | |

Maximum energy storage per unit volume; given velocity |
p |

Maximum energy storage per unit mass; no failure |
Of/p |

aTo minimize cost, use the above criteria for minimum weight, replacing density p by Cmp, where Cm is the material cost per kg. To minimize energy content, use the above criteria for minimum weight replacing density p by qp where q is the energy content per kg.

bof = failure strength (the yield strength for metals and ductile polymers, the tensile strength for ceramics, glasses and brittle polymers); p = density.

cFor design for infinite fatigue life, replace of by the endurance limit oe.

(e) Strength-limited design: springs, hinges etc for maximum performancea

Function and constraints^ Maximizeb

ELASTIC HINGES

Radius of bend to be minimized (max. flexibility without failure) Of /E COMPRESSION SEALS AND GASKETS

Maximum conformability; limit on contact pressure of /E

and 1/E

### ROTATING DRUMS AND CENTRIFUGES

Maximum angular velocity; radius fixed; wall thickness free oy/p aTo minimize cost, use the above criteria for minimum weight, replacing density p by Cmp, where Cm is the material cost per kg. To minimize energy content, use the above criteria for minimum weight replacing density p by qp where q is the energy content per kg.

bOf = failure (the yield strength for metals and ductile polymers, the tensile strength for ceramics, glasses and brittle polymers); H = hardness; p = density.

cFor design for infinite fatigue life, replace of by the endurance limit oe.

(f) Vibration-limited design

Function and constraints^ Maximize"

TIES, COLUMNS

Maximum longitudinal vibration frequencies E/p BEAMS

Maximum flexural vibration frequencies E1/2/p PANELS

Maximum flexural vibration frequencies E1/3/p TIES, COLUMNS, BEAMS, PANELS

Minimum longitudinal excitation from external drivers, ties ^E/p

Minimum flexural excitation from external drivers, beams iqEl/1 /p Minimum flexural excitation from external drives, panels aTo minimize cost, use the above criteria for minimum weight, replacing density p by Cmp, where Cm is the material cost per kg. To minimize energy content, use the above criteria for minimum weight replacing density p by qp where q is the energy content per kg.

bof = failure (the yield strength for metals and ductile polymers, the tensile strength for ceramics, glasses and brittle polymers); ^ = damping coefficient; p = density.

cFor design for infinite fatigue life, replace of by the endurance limit oe.

(g) Thermal and thermo-mechanical Design

### Function and constraints Maximizea

THERMAL INSULATION MATERIALS Minimum heat flux at steady state; thickness specified Minimum temp rise in specified time; thickness specified Minimize total energy consumed in thermal cycle (kilns, etc.)

THERMAL STORAGE MATERIALS

Maximum energy stored/unit material cost (storage heaters)

Maximize energy stored for given temperature rise and time

PRECISION DEVICES

Minimize thermal distortion for given heat flux

THERMAL SHOCK RESISTANCE

Maximum change in surface temperature; no failure

Function and constraints Maximizea

HEAT SINKS

Maximum heat flux per unit volume; expansion limited X/Aa

Maximum heat flux per unit mass; expansion limited X/pAa

HEAT EXCHANGERS (pressure-limited)

Maximum heat flux per unit area; no failure under Ap Xof

Maximum heat flux per unit mass; no failure under Ap Xof/p

= thermal conductivity; a = thermal diffusivity; Cp = specific heat capacity; Cm = material cost / kg; Tmax = maximum service temperature; a = thermal expansion coeff; E = Young's modulus; p = density; of = failure strength (the yield strength for metals and ductile polymers, the tensile strength for ceramics, glass and brittle polymers).

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