Elastic modulus

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An elastic modulus, or modulus of elasticity, is a number that measures an object or substance's resistance to being deformed elastically (i.e., non-permanently) when a force is applied to it. The elastic modulus of an object is defined as the slope of its stress–strain curve in the elastic deformation region:[1] A stiffer material will have a higher elastic modulus. An elastic modulus has the form

\lambda \ \stackrel{\text{def}}{=}\  \frac {\text{stress}} {\text{strain}}

where stress is the restoring force caused by the deformation divided by the area to which the force is applied and strain is the ratio of the change in some length parameter caused by the deformation to the original value of the length parameter. If stress is measured in pascals, then since strain is a dimensionless quantity, the units of λ will be pascals as well.[2]

Since the strain equals unity for an object whose length has doubled, the elastic modulus equals the stress induced in the material by a doubling of length. While this scenario is not generally realistic because most materials will fail before reaching it, it gives heuristic guidance, because small fractions of the defining load will operate in exactly the same ratio. Thus, for steel with a Young's modulus of 30 million psi, a 30 thousand psi load will elongate a 1 inch bar by one thousandth of an inch; similarly, for metric units, a load of one-thousandth of the modulus (now measured in gigapascals) will change the length of a one-meter rod by a millimeter.

Specifying how stress and strain are to be measured, including directions, allows for many types of elastic moduli to be defined. The three primary ones are:

Three other elastic moduli are Axial Modulus, Lamé's first parameter, and P-wave modulus.

Homogeneous and isotropic (similar in all directions) materials (solids) have their (linear) elastic properties fully described by two elastic moduli, and one may choose any pair. Given a pair of elastic moduli, all other elastic moduli can be calculated according to formulas in the table below at the end of page.

Inviscid fluids are special in that they cannot support shear stress, meaning that the shear modulus is always zero. This also implies that Young's modulus is always zero.

See also[edit]

References[edit]

  1. ^ Askeland, Donald R.; Phulé, Pradeep P. (2006). The science and engineering of materials (5th ed.). Cengage Learning. p. 198. ISBN 978-0-534-55396-8. 
  2. ^ Beer, Ferdinand P.; Johnston, E. Russell; Dewolf, John; Mazurek, David (2009). Mechanics of Materials. McGraw Hill. p. 56. ISBN 978-0-07-015389-9. 

Further reading[edit]

Conversion formulas
Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these, thus given any two, any other of the elastic moduli can be calculated according to these formulas.
K=\,E=\, \lambda=\,G=\, \nu=\,M=\,Notes
(K,\,E)KE\tfrac{3K(3K-E)}{9K-E}\tfrac{3KE}{9K-E}\tfrac{3K-E}{6K}\tfrac{3K(3K+E)}{9K-E}
(K,\,\lambda)K\tfrac{9K(K-\lambda)}{3K-\lambda}\lambda\tfrac{3(K-\lambda)}{2}\tfrac{\lambda}{3K-\lambda}3K-2\lambda\,
(K,\,G)K\tfrac{9KG}{3K+G}K-\tfrac{2G}{3}G\tfrac{3K-2G}{2(3K+G)}K+\tfrac{4G}{3}
(K,\,\nu)K3K(1-2\nu)\,\tfrac{3K\nu}{1+\nu}\tfrac{3K(1-2\nu)}{2(1+\nu)}\nu\tfrac{3K(1-\nu)}{1+\nu}
(K,\,M)K\tfrac{9K(M-K)}{3K+M}\tfrac{3K-M}{2}\tfrac{3(M-K)}{4}\tfrac{3K-M}{3K+M}M
(E,\,\lambda)\tfrac{E + 3\lambda + R}{6}E\lambda\tfrac{E-3\lambda+R}{4}\tfrac{2\lambda}{E+\lambda+R}\tfrac{E-\lambda+R}{2}R=\sqrt{E^2+9\lambda^2 + 2E\lambda}
(E,\,G)\tfrac{EG}{3(3G-E)}E\tfrac{G(E-2G)}{3G-E}G\tfrac{E}{2G}-1\tfrac{G(4G-E)}{3G-E}
(E,\,\nu)\tfrac{E}{3(1-2\nu)}E\tfrac{E\nu}{(1+\nu)(1-2\nu)}\tfrac{E}{2(1+\nu)}\nu\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}
(E,\,M)\tfrac{3M-E+S}{6}E\tfrac{M-E+S}{4}\tfrac{3M+E-S}{8}\tfrac{E-M+S}{4M}M

S=\pm\sqrt{E^2+9M^2-10EM}

There are two valid solutions.
The plus sign leads to \nu\geq 0.
The minus sign leads to \nu\leq 0.

(\lambda,\,G)\lambda+ \tfrac{2G}{3}\tfrac{G(3\lambda + 2G)}{\lambda + G}\lambdaG\tfrac{\lambda}{2(\lambda + G)}\lambda+2G\,
(\lambda,\,\nu)\tfrac{\lambda(1+\nu)}{3\nu}\tfrac{\lambda(1+\nu)(1-2\nu)}{\nu}\lambda\tfrac{\lambda(1-2\nu)}{2\nu}\nu\tfrac{\lambda(1-\nu)}{\nu}Cannot be used when \nu=0 \Leftrightarrow \lambda=0
(\lambda,\,M)\tfrac{M + 2\lambda}{3}\tfrac{(M-\lambda)(M+2\lambda)}{M+\lambda}\lambda\tfrac{M-\lambda}{2}\tfrac{\lambda}{M+\lambda}M
(G,\,\nu)\tfrac{2G(1+\nu)}{3(1-2\nu)}2G(1+\nu)\,\tfrac{2 G \nu}{1-2\nu}G\nu\tfrac{2G(1-\nu)}{1-2\nu}
(G,\,M)M - \tfrac{4G}{3}\tfrac{G(3M-4G)}{M-G}M - 2G\,G\tfrac{M - 2G}{2M - 2G}M
(\nu,\,M)\tfrac{M(1+\nu)}{3(1-\nu)}\tfrac{M(1+\nu)(1-2\nu)}{1-\nu}\tfrac{M \nu}{1-\nu}\tfrac{M(1-2\nu)}{2(1-\nu)}\nuM