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:
specificmodulus=E/\rho
where
E
\rho
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 (tensile) 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 (rather than stiffness) issues due to effects of stress risers.
Specific modulus is not to be confused with specific strength, a term that compares strength to density.
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 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.
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.
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 (area moment of inertia) of the beam.
Comparing the list of area moments of inertia with formulas for area gives the appropriate relationship for beams of various configurations.
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.
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.[1] [2] [3]
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.
| Material | Young's modulus (GPa) | Density (g/cm3) | Young's modulus per density; specific stiffness (106 m2s−2) | Young's modulus per density squared (103 m5kg−1s−2) | Young's modulus per density cubed (m8kg−2s−2) | Reference | |
|---|---|---|---|---|---|---|---|
| Latex foam, low density, 10% compression[4] | |||||||
| Reversible Assembled Cellular Composite Materials | [5] [6] | ||||||
| Self Reprogrammable Mechanical Metamaterials | [7] [8] | ||||||
| Latex foam, low density, 40% compression[9] | |||||||
| Latex foam, high density, 10% compression[10] | |||||||
| Latex foam, high density, 40% compression[11] | |||||||
| Silica aerogel, medium density[12] | |||||||
| Rubber (small strain) | ±0.045 | ±0.145[13] | ±0.051 | ±0.05655 | ±0.0621 | ||
| Expanded polystrene (EPS) foam, low density (1lb/ft3)[14] | |||||||
| Silica aerogel, high density[15] | |||||||
| Expanded polystrene (EPS) foam, medium density (3lb/ft3)[16] | |||||||
| Low-density polyethylene | ±0.015 | ±0.005 | ±0.005 | ±0.015 | |||
| PTFE (Teflon) | |||||||
| Duocel aluminum foam, 8% density[17] | |||||||
| Extruded polystrene (XPS) foam, medium density (Foamular 400)[18] [19] | |||||||
| Extruded polystrene (XPS) foam, high density (Foamular 1000)[20] [21] | |||||||
| HDPE | [22] | ||||||
| Duocel copper foam, 8% density[23] | |||||||
| Polypropylene[24] | ±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 | [25] | ||||||
| Titanium foam, low density[26] | |||||||
| Titanium foam, high density[27] | |||||||
| Foam glass[28] | |||||||
| Copper (Cu) | |||||||
| Brass and bronze | ±12.5 | ±0.165 | ±2.0 | ±0.25 | ±0.03 | ||
| Zinc (Zn) | |||||||
| Oak wood (along grain) | ±0.17[29] | ±3.5 | ±9.5 | ±20.0 | |||
| Concrete (under compression) | ±10 | ±4 | ±1.75 | ±0.7 | |||
| Glass-reinforced plastic[30] [31] [32] | ±14.45 | ±8 | ±4.35 | ±2.5 | |||
| Pine wood | ±0.155 | ±6 | ±26 | ±89 | |||
| Balsa, low density (4.4 lb/ft3)[33] | |||||||
| Tungsten (W) | |||||||
| Sitka spruce green[34] [35] [36] | ±0.7 | ±2 | ±5 | ±13 | |||
| Osmium (Os) | |||||||
| Balsa, medium density (10 lb/ft3)[37] | |||||||
| Steel | ±0.15 | ±0.5 | ±0.1 | ±0.02 | |||
| Titanium alloys | ±7.5 | ±2 | ±0.35 | ±0.08 | |||
| Balsa, high density (16 lb/ft3)[38] | |||||||
| Wrought iron | ±10 | ±0.2 | ±2 | ±0.35 | ±0.055 | ||
| Magnesium metal (Mg) | |||||||
| Sitka spruce dry | ±0.8 | ±2 | ±5 | ±12 | |||
| Macor machineable glass-ceramic[39] | |||||||
| Cordierite[40] | |||||||
| Glass | ±20 | ±0.2[41] | ±10 | ±4.8 | ±2.1 | ||
| Tooth enamel (largely calcium phosphate) | [42] | ||||||
| E-Glass fiber[43] [44] | |||||||
| Molybdenum (Mo) | |||||||
| Basalt fiber | |||||||
| Zirconia[45] | |||||||
| Tungsten carbide (WC) | ±100 | ±6.5 | ±0.4 | ±0.025 | |||
| S-Glass fiber[46] | |||||||
| Flax fiber[47] [48] [49] [50] | ±34 | ±0.15 | ±29.35 | ±25 | ±21 | ||
| single-crystal Yttrium iron garnet (YIG) | [51] | ||||||
| Kevlar 29[52] (tensile only[53]) | |||||||
| Steatite L-5[54] | |||||||
| Mullite[55] | |||||||
| Dyneema SK25 Ultra-high-molecular-weight polyethylene (tensile only)[56] | |||||||
| Beryllium, 30% porosity[57] | |||||||
| Kevlar 49 (tensile only) | |||||||
| Silicon[58] | |||||||
| Alumina fiber (Al2O3)[59] [60] | ±0.315 | ±7 | ±4 | ±1.74 | |||
| Syalon 501 Silicon nitride[61] | |||||||
| Sapphire[62] | |||||||
| Alumina[63] | |||||||
| Carbon fiber reinforced plastic (70:30 fibre:matrix, unidirectional, along grain)[64] | |||||||
| Dyneema SK78/Honeywell Spectra 2000 UHMWPE (tensile only)[65] | ±11 | ±11 | ±12 | ±12 | |||
| Silicon carbide (SiC) | |||||||
| Beryllium (Be) | |||||||
| Boron fiber[66] | |||||||
| Boron nitride[67] | |||||||
| Diamond (C) | |||||||
| Dupont E130 carbon fiber[68] |
| Material | Young's modulus (GPa) | Density (g/cm3) | Young's modulus per density; specific stiffness (106 m2s−2) | Young's modulus per density squared (103 m5kg−1s−2) | Young's modulus per density cubed (m8kg−2s−2) | |
|---|---|---|---|---|---|---|
| Applewood or wild apple (Pyrus malus) | ||||||
| Ash, black (Fraxinus nigra) | ||||||
| Ash, blue (quadrangulata) | ||||||
| Ash, green (Fraxinus pennsylvanica lanceolata) | ||||||
| Ash, white (Fraxinus americana) | ||||||
| Aspen (Populus tremuloides) | ||||||
| Aspen, large tooth (Populus grandidentata) | ||||||
| Basswood (Tilia glabra or Tilia americanus) | ||||||
| Beech (Fagus grandifolia or Fagus americana) | ||||||
| Beech, blue (Carpinus caroliniana) | ||||||
| Birch, gray (Betula populifolia) | ||||||
| Birch, paper (Betula papyrifera) | ||||||
| Birch, sweet (Betula lenta) | ||||||
| Buckeye, yellow (Aesculus octandra) | ||||||
| Butternut (Juglans cinerea) | ||||||
| Cedar, eastern red (Juniperus virginiana) | ||||||
| Cedar, northern white (Thuja occidentalis) | ||||||
| Cedar, southern white (Chamaecyparis thvoides) | ||||||
| Cedar, western red (Thuja plicata) | ||||||
| Cherry, black (Prunus serotina) | ||||||
| Cherry, wild red (Prunus pennsylvanica) | ||||||
| Chestnut (Castanea dentata) | ||||||
| Cottonwood, eastern (Populus deltoides) | ||||||
| Cypress, southern (Taxodium distichum) | ||||||
| Dogwood (flowering) (Cornus Florida) | ||||||
| Douglas fir (coast type) (Pseudotsuga taxifolia) | ||||||
| Douglas fir (mountain type) (Pseudotsuga taxifolia) | ||||||
| Ebony, Andaman marble-wood (India) (Diospyros kursii) | ||||||
| Ebony, Ebè marbre (Mauritius, E. Africa) (Diospyros melanida) | ||||||
| Elm, American (Ulmus americana) | ||||||
| Elm, rock (Ulmus racemosa or Ulmus thomasi) | ||||||
| Elm, slippery (Ulmus fulva or pubescens) | ||||||
| Eucalyptus, Karri (W. Australia) (Eucalyptus diversicolor) | ||||||
| Eucalyptus, Mahogany (New South Wales) (Eucalyptus hemilampra) | ||||||
| Eucalyptus, West Australian mahogany (Eucalyptus marginata) | ||||||
| Fir, balsam (Abies balsamea) | ||||||
| Fir, silver (Abies amabilis) | ||||||
| Gum, black (Nyssa sylvatica) | ||||||
| Gum, blue (Eucalyptus globulus) | ||||||
| Gum, red (Liquidambar styraciflua) | ||||||
| Gum, tupelo (Nyssa aquatica) | ||||||
| Hemlock eastern (Tsuga canadensis) | ||||||
| Hemlock, mountain (Tsuga martensiana) | ||||||
| Hemlock, western (Tsuga heterophylla) | ||||||
| Hickory, bigleaf shagbark (Hicoria laciniosa) | ||||||
| Hickory, mockernut (Hicoria alba) | ||||||
| Hickory, pignut (Hicoria glabra) | ||||||
| Hickory, shagbark (Hicoria ovata) | ||||||
| Hornbeam (Ostrya virginiana) | ||||||
| Ironwood, black (Rhamnidium ferreum) | ±1.64 | ±2.78 | ±3.56 | |||
| Larch, western (Larix occidentalis) | ||||||
| Locust, black or yellow (Robinia pseudacacia) | ||||||
| Locust honey (Gleditsia triacanthos) | ||||||
| Magnolia, cucumber (Magnolia acuminata) | ||||||
| Mahogany (W. Africa) (Khaya ivorensis) | ||||||
| Mahogany (E. India) (Swietenia macrophylla) | ||||||
| Mahogany (E. India) (Swietenia mahogani) | ||||||
| Maple, black (Acer nigrum) | ||||||
| Maple, red (Acer rubrum) | ||||||
| Maple, silver (Acer saccharinum) | ||||||
| Maple, sugar (Acer saccharum) | ||||||
| Oak, black (Quercus velutina) | ||||||
| Oak, bur (Quercus macrocarpa) | ||||||
| Oak, canyon live (Quercus chrysolepis) | ||||||
| Oak, laurel (Quercus Montana) | ||||||
| Oak, live (Quercus virginiana) | ||||||
| Oak, post (Quercus stellata or Quercus minor) | ||||||
| Oak, red (Quercus borealis) | ||||||
| Oak, swamp chestnut (Quercus Montana (Quercus prinus)) | ||||||
| Oak swamp white (Quercus bicolor or Quercus platanoides) | ||||||
| Oak, white (Quercus alba) | ||||||
| Paulownia (P. tomentosa) | ||||||
| Persimmon (Diospyros virginiana) | ||||||
| Pine, eastern white (Pinus strobus) | ||||||
| Pine, jack (Pinus banksiana or Pinus divericata) | ||||||
| Pine, loblolly (Pinus taeda) | ||||||
| Pine, longleaf (Pinus palustris) | ||||||
| Pine, pitch (Pinus rigida) | ||||||
| Pine, red (Pinus resinosa) | ||||||
| Pine, shortleaf (Pinus echinata) | ||||||
| Poplar, balsam (Populus balsamifera or Populus candicans) | ||||||
| Poplar, yellow (Liriodendron tulipifera) | ||||||
| Redwood (Sequoia sempervirens) | ||||||
| Sassafras (Sassafras uariafolium) | ||||||
| Satinwood (Ceylon) (Chloroxylon swietenia) | ||||||
| Sourwood (Oxydendrum arboreum) | ||||||
| Spruce, black (Picea mariana) | ||||||
| Spruce, red (Picea rubra or Picea rubens) | ||||||
| Spruce, white (Picea glauca) | ||||||
| Sycamore (Platanus occidentalis) | ||||||
| Tamarack (Larix laricina or Larix americana) | ||||||
| Teak (India) (Tectona grandis) | ||||||
| Walnut, black (Juglans nigra) | ||||||
| Willow, black (Salix nigra) |
| Material | Young's modulus (GPa) | Density (g/cm3) | Young's modulus per density; specific stiffness (106 m2s−2) | Young's modulus per density squared (103 m5kg−1s−2) | Young's modulus per density cubed (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 |