Specific modulus

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Template:Short description 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:

specific modulus=E/ρ

where E is the elastic modulus and ρ 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 (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

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 (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

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.[1][2][3]

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

Specific stiffness of the full range of materials
Specific stiffness of materials within the range 0.9–5.0Template:Nbspg/cm3 density and 10–1300Template:NbspGPa stiffness
Approximate specific stiffness for various materials. No attempt is made to correct for materials whose stiffness varies with their density.
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] Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
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Approximate specific stiffness for various species of wood[53]
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)
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Specific stiffness of the elements[54][55]
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)
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Cadmium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Neodymium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Hafnium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Lanthanum Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Promethium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Thorium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Samarium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Lutetium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Terbium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Tin Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Tellurium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Gadolinium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Dysprosium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Holmium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Erbium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Platinum Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Thulium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Silver Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Antimony Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Lithium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Palladium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Zirconium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Sodium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Uranium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Tantalum Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Niobium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Calcium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Yttrium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Copper Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Zinc Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Silicon Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Vanadium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Tungsten Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Rhenium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Rhodium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Nickel Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Iridium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Cobalt Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Scandium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Titanium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Magnesium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Aluminum Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Manganese Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Iron Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Molybdenum Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Ruthenium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Chromium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts
Beryllium Template:Nts Template:Nts Template:Nts Template:Nts Template:Nts

See also

References

  1. ↑ Template:Cite web
  2. ↑ Template:Cite web
  3. ↑ Template:Cite book
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