Specific modulus

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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 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.

Applications[edit]

Specific stiffness in tension[edit]

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[edit]

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[edit]

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[edit]

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.

Beam's cross-sectional area increases in two dimensions[edit]

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[edit]

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]

Stiffness versus strength in bending[edit]

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[edit]

Approximate specific stiffness for various materials. No attempt is made to correct for materials whose stiffness varies with their density.
Material Young's modulus in GPa Density in g/cm3 Young's modulus over density in 106 m2s−2 Young's modulus over density squared in 103 m5kg−1s−2 Young's modulus over density cubed in m8kg−2s−2
Rubber (small strain) 0.055±0.045 1.055±0.145[4] 0.059±0.051 0.06345±0.05655 0.0679±0.0621
Low density polyethylene 0.2 0.925±0.015 0.215±0.005 0.235±0.005 0.255±0.015
PTFE (Teflon) 0.5 2.2 0.23 0.10 0.047
HDPE 0.8 0.95[5] 0.84 0.89 0.93
Polypropylene [6] 1.2±0.3 0.9 1.33±0.33 1.48±0.37 1.65±0.41
Polyethylene terephthalate 2.35±0.35 1.4125±0.0425 1.7±0.3 1.17±0.23 0.875±0.225
Nylon 3.0±1.0 1.15 2.6±0.9 2.25±0.75 1.95±0.65
Polystyrene 3.25±0.25 1.05 3.1±0.2 2.95±0.25 2.8±0.2
Biaxially-oriented Polypropylene[7] 3.2±1.0 0.9 3.56±1.11 3.95±1.23 4.39±1.37
Medium-density fibreboard 4 0.75[8] 5.3 7.1 9.5
Zinc (Zn) 108 7.14 15 2.1 0.29
Copper (Cu) 117 8.94 13 1.5 0.16
Brass and bronze 112.5±12.5 8.565±0.165 13.0±2.0 1.55±0.25 0.18±0.03
Oak wood (along grain) 11 0.76±0.17[9] 15.5±3.5 22.5±9.5 34.0±20.0
Concrete (under compression) 40±10 2.4 17±4 6.95±1.75 2.9±0.7
Average lightweight Balsa[10] 2.3 0.11 21 190 1,700
"Unreal Stiff" lightweight Balsa[11][12][13] 1.67 0.08 21 260 3,200
Tungsten (W) 400 19.25 21 1.1 0.056
Osmium (Os) 550 22.59 24 1.1 0.048
Wrought iron 200±10 7.7±0.2 26±2 3.35±0.35 0.445±0.055
Glass-reinforced plastic[14][15][16] 31.65±14.45 1.8 18±8 9.65±4.35 5.4±2.5
Pine wood 8.963 0.505±0.155[17] 20±6 47±26 120±89
Sitka spruce green[18][19][20] 8.7±0.7 0.37 23.5±2 64±5 172±13
Steel 200 7.9±0.15 25±0.5 3.2±0.1 0.41±0.02
Magnesium metal (Mg) 45 1.738 26 15 8.6
Aluminium 69 2.7 26 9.5 3.5
Titanium alloys 112.5±7.5 4.5 25±2 5.55±0.35 1.23±0.08
Sitka spruce dry[21][22][23] 10.4±0.8 0.4 26±2 65±5 162±12
Tooth enamel (largely calcium phosphate) 83 2.8[24] 30 11 3.8
E-Glass fiber[25][26] 81 2.62 31 12 4.5
Molybdenum (Mo) 329 10.28 32 3.1 0.30
Basalt fiber 89 2.7 33 12 4.5
S-Glass fiber[27][28] 89 2.5 36 14 5.7
Glass 70±20 2.6±0.2[29] 28±10 11.2±4.8 4.4±2.1
single-crystal Yttrium iron garnet (YIG) 200 5.17[30] 39 7.5 1.4
Tungsten carbide (WC) 550±100 15.8 34.5±6.5 2.2±0.4 0.135±0.025
Flax fiber[31][32][33][34] 45±34 1.35±0.15 36.65±29.35 30±25 25±21
Jute fiber (tension)[35] 55.5 1.3 42.7 32.8 25.3
Kevlar 29[36] (tensile only[37]) 70.5 1.44 49 34 24
Dyneema SK25 Ultra-high-molecular-weight polyethylene (tensile only)[38] 52 0.97 54 55 57
Kevlar 49[39] (tensile only[40]) 112.4 1.44 78 54 38
Silicon[41] 185 2.329 79 34 15
Alumina fiber (Al2O3)[42][43][44] 300 3.595±0.315 84±7 24±4 6.76±1.74
Carbon fiber reinforced plastic (70/30 fibre/matrix, unidirectional, along grain)[45] 181 1.6 113 71 44
Dyneema SK78/Honeywell Spectra 2000 Ultra-high-molecular-weight polyethylene (tensile only)[46][47] 121±11 0.97 125±11 128±12 132±12
Silicon carbide (SiC) 450 3.21 140 44 14
Beryllium (Be) 287 1.85 155 84 45
Boron fiber[48] 400 2.54 157 62 24
Diamond (C) 1,220 3.53 346 98 28
Dupont E130 carbon fiber[49] 896 2.15 417 194 90
Approximate (to 6 significant digits) specific stiffness for various species of wood[50]
Species Young's modulus in GPa Density in g/cm3 Young's modulus over density in 106 m2s−2 Young's modulus over density squared in 103 m5kg−1s−2 Young's modulus over density cubed in m8kg−2s−2
Applewood or wild apple (Pyrus malus ) 8.76715 0.745 11.768 15.7959 21.2026
Ash, black (Fraxinus nigra ) 11.0423 0.526 20.9929 39.9105 75.8755
Ash, blue (quadrangulata ) 9.64974 0.603 16.0029 26.5388 44.0113
Ash, green (Fraxinus pennsylvanica lanceolata ) 11.4738 0.610 18.8095 30.8352 50.5495
Ash, white (Fraxinus americana ) 12.2485 0.638 19.1983 30.0914 47.1651
Aspen (Populus tremuloides ) 8.21797 0.401 20.4937 51.1065 127.448
Aspen, large tooth (PopuIus grandidentata ) 9.76742 0.412 23.7073 57.5421 139.665
Basswood (Tilia glabra or Tilia americanus ) 10.091 0.398 25.3544 63.7045 160.061
Beech (Fagus grandifolia or Fagus americana ) 11.5718 0.655 17.6669 26.9724 41.1793
Beech, blue (Carpinus caroliniana ) 7.3746 0.717 10.2854 14.345 20.007
Birch, gray (Betula populifolia ) 7.8159 0.552 14.1592 25.6508 46.4688
Birch, paper (Betula papyrifera ) 10.9736 0.600 18.2894 30.4823 50.8039
Birch, sweet (Betula lenta ) 14.9061 0.714 20.8769 29.2394 40.9515
Buckeye, yellow (Aesculus octandra ) 8.12971 0.383 21.2264 55.4214 144.703
Butternut (Juglans cinerea ) 8.13952 0.404 20.1473 49.8696 123.44
Cedar, eastern red (Juniperus virginiana ) 6.00167 0.492 12.1985 24.7937 50.3938
Cedar, northern white (Thuja occidentalis ) 5.57018 0.315 17.6831 56.1368 178.212
Cedar, southern white (Chamaecyparis thvoides ) 6.42336 0.352 18.2482 51.8414 147.277
Cedar, western red (Thuja plicata ) 8.03165 0.344 23.3478 67.8715 197.301
Cherry, black (Prunus serotina ) 10.2578 0.534 19.2093 35.9724 67.3641
Cherry, wild red (Prunus pennsylvanica ) 8.74753 0.425 20.5824 48.4292 113.951
Chestnut (Castanea dentata ) 8.53179 0.454 18.7925 41.3931 91.1743
Cottonwood, eastern (Populus deltoides ) 9.53206 0.433 22.014 50.8407 117.415
Cypress, southern (Taxodium distichum ) 9.90472 0.482 20.5492 42.6332 88.4506
Dogwood (flowering) (Cornus florida ) 10.6402 0.796 13.3671 16.7928 21.0965
Douglas fir (coast type) (Pseudotsuga taxifolia ) 13.3076 0.512 25.9915 50.7646 99.1495
Douglas fir (mountain type) (Pseudotsuga taxifolia ) 9.62032 0.446 21.5702 48.3637 108.439
Ebony, Andaman marble-wood (India) (Diospyros kursii ) 12.4544 0.978 12.7346 13.0211 13.314
Ebony, Ebè marbre (Maritius, E. Africa) (Diospyros melanida ) 9.8753 0.768 12.8585 16.7428 21.8005
Elm, American (Ulmus americana ) 9.2967 0.554 16.7811 30.2907 54.6764
Elm, rock (Ulmus racemosa or Ulmus thomasi ) 10.65 0.658 16.1854 24.5979 37.3829
Elm, slippery (Ulmus fulva or pubescens ) 10.297 0.568 18.1285 31.9164 56.1908
Eucalyptus, Karri (W. Australia) (Eucalyptus diversicolor ) 18.4855 0.829 22.2986 26.8982 32.4465
Eucalyptus, Mahogany (New South Wales)(Eucalyptus hemilampra) 15.7691 1.058 14.9046 14.0875 13.3153
Eucalyptus, West Australian mahogany (Eucalyptus marginata ) 14.3373 0.787 18.2177 23.1483 29.4133
Fir, balsam (Abies balsamea) 8.62005 0.414 20.8214 50.2932 121.481
Fir, silver (Abies amabilis ) 10.552 0.415 25.4264 61.2684 147.635
Gum, black (Nyssa sylvatica ) 8.22778 0.552 14.9054 27.0025 48.9176
Gum, blue (Eucalyptus globulus ) 16.5046 0.796 20.7344 26.0483 32.7239
Gum, red (Liquidambar styraciflua ) 10.2479 0.530 19.3358 36.4826 68.835
Gum, tupelo (Nyssa aquatica ) 8.71811 0.524 16.6376 31.7512 60.5939
Hemlock eastern (Tsuga canadensis ) 8.29643 0.431 19.2492 44.6618 103.624
Hemlock, mountain (Tsuga martensiana ) 7.8159 0.480 16.2831 33.9232 70.6733
Hemlock, western (Tsuga heterophylla ) 9.95375 0.432 23.0411 53.3359 123.463
Hickory, bigleaf shagbark (Hicoria laciniosa ) 13.0919 0.809 16.1828 20.0034 24.7261
Hickory, mockernut (Hicoria alba ) 15.3964 0.820 18.7761 22.8977 27.9241
Hickory, pignut (Hicoria glabra ) 15.7201 0.820 19.1708 23.379 28.511
Hickory, shagbark (Hicoria ovata ) 14.9551 0.836 17.8889 21.3982 25.596
Hornbeam (Ostrya virginiana ) 11.7582 0.762 15.4307 20.2502 26.5751
Ironwood, black (Rhamnidium ferreum ) 20.594 −-2.377000 17.48±1.64 14.97±2.78 12.93±3.56
Larch, western (Larix occidentalis ) 11.6503 0.587 19.8472 33.8112 57.6
Locust, black or yellow (Robinia pseudacacia ) 14.2 0.708 20.0565 28.3284 40.0119
Locust honey (Gleditsia triacanthos ) 11.4247 0.666 17.1543 25.7572 38.6744
Magnolia, cucumber (Magnolia acuminata ) 12.5133 0.516 24.2506 46.9972 91.0798
Mahogany (W. Africa) (Khaya ivorensis ) 10.5814 0.668 15.8404 23.7131 35.4987
Mahogany (E. India) (Swietenia macrophylla ) 8.01203 0.54 14.8371 27.4761 50.8817
Mahogany (E. India) (Swietenia mahogani ) 8.72792 0.54 16.1628 29.9311 55.428
Maple, black (Acer nigrum ) 11.1894 0.620 18.0474 29.1087 46.9495
Maple, red (Acer rubrum ) 11.3267 0.546 20.7448 37.9942 69.5865
Maple, silver (Acer saccharinum ) 7.89435 0.506 15.6015 30.833 60.9347
Maple, sugar (Acer saccharum ) 12.6506 0.676 18.7139 27.6832 40.9515
Oak, black (Quercus velutina ) 11.3071 0.669 16.9014 25.2637 37.7634
Oak, bur (Quercus macrocarpa ) 7.09021 0.671 10.5666 15.7476 23.4688
Oak, canyon live (Quercus chrysolepis ) 11.2678 0.838 13.4461 16.0455 19.1473
Oak, laurel (Quercus montana ) 10.9246 0.674 16.2086 24.0484 35.6801
Oak, live (Quercus virginiana ) 13.543 0.977 13.8618 14.1881 14.5221
Oak, post (Quercus stellata or Quercus minor ) 10.4245 0.738 14.1253 19.14 25.9349
Oak, red (Quercus borealis ) 12.4937 0.657 19.0162 28.9441 44.0549
Oak, swamp chestnut (Quercus montana (Quercus prinus) ) 12.2289 0.756 16.1758 21.3965 28.3023
Oak swamp white (Quercus bicolor or Quercus platanoides ) 14.1804 0.792 17.9046 22.6068 28.5439
Oak, white (Quercus alba ) 12.2681 0.710 17.279 24.3367 34.277
Paulownia (P. tomentosa) 6.894 0.274 27.9109 91.8269 335.134
Persimmon (Diospyros virginiana ) 14.151 0.776 18.2358 23.4998 30.2832
Pine, eastern white (Pinus strobus ) 8.80637 0.373 23.6096 63.2964 169.696
Pine, jack (Pinus banksiana or Pinus divericata ) 8.51217 0.461 18.4646 40.0533 86.8836
Pine, loblolly (Pinus taeda ) 13.2782 0.593 22.3916 37.7598 63.6759
Pine, longleaf (Pinus palustris ) 14.1706 0.638 22.211 34.8135 54.5665
Pine, pitch (Pinus rigida ) 9.46342 0.542 17.4602 32.2144 59.4361
Pine, red (Pinus resinosa ) 12.3956 0.507 24.4489 48.2227 95.1139
Pine, shortleaf (Pinus echinata ) 13.1899 0.584 22.5855 38.6738 66.2223
Poplar, balsam (Populus balsamifera or Populus candicans ) 7.02156 0.331 21.2132 64.0881 193.62
Poplar, yellow (Liriodendron tulipifera ) 10.3754 0.427 24.2984 56.905 133.267
Redwood (Sequoia sempervirens ) 9.39477 0.436 21.5476 49.4212 113.351
Sassafras (Sassafras uariafolium ) 7.74725 0.473 16.379 34.6278 73.209
Satinwood (Ceylon) (Chloroxylon swietenia ) 10.7971 1.031 10.4725 10.1576 9.85217
Sourwood (Oxydendrum arboreum ) 10.6206 0.593 17.91 30.2023 50.9313
Spruce, black (Picea mariana ) 10.4833 0.428 24.4937 57.2283 133.711
Spruce, red (Picea rubra or Picea rubens ) 10.5029 0.413 25.4308 61.5758 149.094
Spruce, white (Picea glauca ) 9.81646 0.431 22.776 52.8446 122.609
Sycamore (Platanus occidentalis ) 9.82626 0.539 18.2305 33.8229 62.7512
Tamarack (Larix laricina or Larix americana ) 11.3169 0.558 20.2811 36.3461 65.1364
Teak (India) (Tectona grandis ) 11.7189 0.5892 19.8896 33.7569 57.2928
Walnut, black (Juglans nigra ) 11.6209 0.562 20.6777 36.7931 65.4682
Willow, black (Salix nigra ) 5.03081 0.408 12.3304 30.2216 74.0726

See also[edit]

References[edit]

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