Specific modulus


Specific modulus is a materials property consisting of the elastic modulus per mass density of a material. It is also known as the stiffness to weight ratio or specific stiffness. High specific modulus materials find wide application in aerospace applications where minimum structural weight is required. The dimensional analysis yields units of distance squared per time squared. The equation can be written as:
where is the elastic modulus and is the density.
The utility of specific modulus is to find materials which will produce structures with minimum weight, when the primary design limitation is deflection or physical deformation, rather than load at breaking—this is also known as a "stiffness-driven" structure. Many common structures are stiffness-driven over much of their use, such as airplane wings, bridges, masts, and bicycle frames.
To emphasize the point, consider the issue of choosing a material for building an airplane. Aluminum seems obvious because it is "lighter" than steel, but steel is stronger than aluminum, so one could imagine using thinner steel components to save weight without sacrificing strength. The problem with this idea is that there would be a significant sacrifice of stiffness, allowing, e.g., wings to flex unacceptably. Because it is stiffness, not tensile strength, that drives this kind of decision for airplanes, we say that they are stiffness-driven.
The connection details of such structures may be more sensitive to strength issues due to effects of stress risers.
Specific modulus is not to be confused with specific strength, a term that compares strength to density.

Applications

Specific stiffness in tension

The use of specific stiffness in tension applications is straightforward. Both stiffness in tension and total mass for a given length are directly proportional to cross-sectional area. Thus performance of a beam in tension will depend on Young's modulus divided by density.

Specific stiffness in buckling and bending

Specific stiffness can be used in the design of beams subject to bending or Euler buckling, since bending and buckling are stiffness-driven. However, the role that density plays changes depending on the problem's constraints.

Beam with fixed dimensions; goal is weight reduction

Examining the formulas for buckling and deflection, we see that the force required to achieve a given deflection or to achieve buckling depends directly on Young's modulus.
Examining the density formula, we see that the mass of a beam depends directly on the density.
Thus if a beam's cross-sectional dimensions are constrained and weight reduction is the primary goal, performance of the beam will depend on Young's modulus divided by density.

Beam with fixed weight; goal is increased stiffness

By contrast, if a beam's weight is fixed, its cross-sectional dimensions are unconstrained, and increased stiffness is the primary goal, the performance of the beam will depend on Young's modulus divided by either density squared or cubed. This is because a beam's overall stiffness, and thus its resistance to Euler buckling when subjected to an axial load and to deflection when subjected to a bending moment, is directly proportional to both the Young's modulus of the beam's material and the second moment of area of the beam.
Comparing the list of area moments of inertia with formulas for area gives the appropriate relationship for beams of various configurations.
Beam's cross-sectional area increases in two dimensions
Consider a beam whose cross-sectional area increases in two dimensions, e.g. a solid round beam or a solid square beam.
By combining the area and density formulas, we can see that the radius of this beam will vary with approximately the inverse of the square of the density for a given mass.
By examining the formulas for area moment of inertia, we can see that the stiffness of this beam will vary approximately as the fourth power of the radius.
Thus the second moment of area will vary approximately as the inverse of the density squared, and performance of the beam will depend on Young's modulus divided by density squared.
Beam's cross-sectional area increases in one dimension
Consider a beam whose cross-sectional area increases in one dimension, e.g. a thin-walled round beam or a rectangular beam whose height but not width is varied.
By combining the area and density formulas, we can see that the radius or height of this beam will vary with approximately the inverse of the density for a given mass.
By examining the formulas for area moment of inertia, we can see that the stiffness of this beam will vary approximately as the third power of the radius or height.
Thus the second moment of area will vary approximately as the inverse of the cube of the density, and performance of the beam will depend on Young's modulus divided by density cubed.
However, caution must be exercised in using this metric. Thin-walled beams are ultimately limited by local buckling and lateral-torsional buckling. These buckling modes depend on material properties other than stiffness and density, so the stiffness-over-density-cubed metric is at best a starting point for analysis. For example, most wood species score better than most metals on this metric, but many metals can be formed into useful beams with much thinner walls than could be achieved with wood, given wood's greater vulnerability to local buckling. The performance of thin-walled beams can also be greatly modified by relatively minor variations in geometry such as flanges and stiffeners.

Stiffness versus strength in bending

Note that the ultimate strength of a beam in bending depends on the ultimate strength of its material and its section modulus, not its stiffness and second moment of area. Its deflection, however, and thus its resistance to Euler buckling, will depend on these two latter values.

Approximate specific stiffness for various materials

MaterialYoung's modulus in GPaDensity in g/cm3Young's modulus over density in 106 m2s−2 Young's modulus over density squared in 103 m5kg−1s−2Young's modulus over density cubed in m8kg−2s−2
Latex foam, low density, 10% compression
Latex foam, low density, 40% compression
Latex foam, high density, 10% compression
Latex foam, high density, 40% compression
Silica aerogel, medium density
Rubber ±0.045±0.145±0.051±0.05655±0.0621
Expanded Polystrene foam, low density
Silica aerogel, high density
Expanded Polystrene foam, medium density
Low-density polyethylene±0.015±0.005±0.005±0.015
PTFE
Duocel aluminum foam, 8% density
Extruded Polystrene foam, medium density
Extruded Polystrene foam, high density
HDPE
Duocel copper foam, 8% density
Polypropylene±0.3±0.33±0.37±0.41
Polyethylene terephthalate±0.35±0.0425±0.3±0.23±0.225
Nylon±1.0±0.9±0.75±0.65
Polystyrene±0.25±0.2±0.25±0.2
Biaxially-oriented Polypropylene±1.0±1.11±1.23±1.37
Medium-density fibreboard
Titanium foam, low density
Titanium foam, high density
Foam glass
Copper
Brass and bronze±12.5±0.165±2.0±0.25±0.03
Zinc
Oak wood ±0.17±3.5±9.5±20.0
Concrete ±10±4±1.75±0.7
Glass-reinforced plastic±14.45±8±4.35±2.5
Pine wood±0.155±6±26±89
Balsa, low density
Tungsten
Sitka spruce green±0.7±2±5±13
Osmium
Balsa, medium density
Steel±0.15±0.5±0.1±0.02
Titanium alloys±7.5±2±0.35±0.08
Balsa, high density
Wrought iron±10±0.2±2±0.35±0.055
Magnesium metal
Aluminium
Sitka spruce dry±0.8±2±5±12
Macor machineable glass-ceramic
Cordierite
Glass±20±0.2±10±4.8±2.1
Tooth enamel
E-Glass fiber
Molybdenum
Basalt fiber
Zirconia
Tungsten carbide ±100±6.5±0.4±0.025
S-Glass fiber
Flax fiber±34±0.15±29.35±25±21
single-crystal Yttrium iron garnet
Jute fiber
Kevlar 29
Steatite L-5
Mullite
Dyneema SK25 Ultra-high-molecular-weight polyethylene
Beryllium, 30% porosity
Kevlar 49
Silicon
Alumina fiber ±0.315±7±4±1.74
Syalon 501 Silicon nitride
Sapphire
Alumina
Carbon fiber reinforced plastic
Dyneema SK78/Honeywell Spectra 2000 Ultra-high-molecular-weight polyethylene ±11±11±12±12
Silicon carbide
Beryllium
Boron fiber
Boron nitride
Diamond
Dupont E130 carbon fiber

SpeciesYoung's modulus in GPaDensity in g/cm3Young's modulus over density in 106 m2s−2 Young's modulus over density squared in 103 m5kg−1s−2Young's modulus over density cubed in m8kg−2s−2
Applewood or wild apple
Ash, black
Ash, blue
Ash, green
Ash, white
Aspen
Aspen, large tooth
Basswood
Beech
Beech, blue
Birch, gray
Birch, paper
Birch, sweet
Buckeye, yellow
Butternut
Cedar, eastern red
Cedar, northern white
Cedar, southern white
Cedar, western red
Cherry, black
Cherry, wild red
Chestnut
Cottonwood, eastern
Cypress, southern
Dogwood
Douglas fir
Douglas fir
Ebony, Andaman marble-wood
Ebony, Ebè marbre
Elm, American
Elm, rock
Elm, slippery
Eucalyptus, Karri
Eucalyptus, Mahogany
Eucalyptus, West Australian mahogany
Fir, balsam
Fir, silver
Gum, black
Gum, blue
Gum, red
Gum, tupelo
Hemlock eastern
Hemlock, mountain
Hemlock, western
Hickory, bigleaf shagbark
Hickory, mockernut
Hickory, pignut
Hickory, shagbark
Hornbeam
Ironwood, black ±1.64±2.78±3.56
Larch, western
Locust, black or yellow
Locust honey
Magnolia, cucumber
Mahogany
Mahogany
Mahogany
Maple, black
Maple, red
Maple, silver
Maple, sugar
Oak, black
Oak, bur
Oak, canyon live
Oak, laurel
Oak, live
Oak, post
Oak, red
Oak, swamp chestnut
Oak swamp white
Oak, white
Paulownia
Persimmon
Pine, eastern white
Pine, jack
Pine, loblolly
Pine, longleaf
Pine, pitch
Pine, red
Pine, shortleaf
Poplar, balsam
Poplar, yellow
Redwood
Sassafras
Satinwood
Sourwood
Spruce, black
Spruce, red
Spruce, white
Sycamore
Tamarack
Teak
Walnut, black
Willow, black

MaterialYoung's modulus in GPaDensity in g/cm3Young's modulus over density in 106 m2s−2 Young's modulus over density squared in 103 m5kg−1s−2Young's modulus over density cubed in m8kg−2s−2
Thallium
Cesium
Arsenic
Lead
Indium
Rubidium
Selenium
Bismuth
Europium
Ytterbium
Barium
Gold
Plutonium
Cerium
Praseodymium
Cadmium
Neodymium
Hafnium
Lanthanum
Promethium
Thorium
Samarium
Lutetium
Terbium
Tin
Tellurium
Gadolinium
Dysprosium
Holmium
Erbium
Platinum
Thulium
Silver
Antimony
Lithium
Palladium
Zirconium
Sodium
Uranium
Tantalum
Niobium
Calcium
Yttrium
Copper
Zinc
Silicon
Vanadium
Tungsten
Rhenium
Rhodium
Nickel
Iridium
Cobalt
Scandium
Titanium
Magnesium
Aluminum
Manganese
Iron
Molybdenum
Ruthenium
Chromium
Beryllium