Elastic modulus

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An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed when a force is applied to it. The elastic modulus of an object is defined as the slope of its stress-strain curve:

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

where λ is the elastic modulus; stress is the force causing the deformation divided by the area to which the force is applied; and strain is the ratio of the change caused by the stress to the original state of the object. Because stress is measured in pascals and strain is a unitless ratio, the units of λ are therefore pascals as well. An alternative definition is that the elastic modulus is the stress required to cause a sample of the material to double in length. This is not literally true for most materials because the value is far greater than the yield stress of the material or the point where elongation becomes nonlinear but some may find this definition more intuitive.

The concept of a constant elastic modulus is dependent on the assumption that the stress-strain curve is always linear. In reality, the curve is only linear within certain limits, because an object stretched or compressed too far will break, and an object under high pressure may undergo processes that will affect the stress-strain curve, such as chemical reactions or buckling.

There are three primary elastic moduli, each describing a different kind of deformation:

The Young's modulus (E) describes tensile elasticity, or the tendency of an object to deform along an axis when opposing forces are applied along that axis; it is defined as the ratio of tensile stress to tensile strain. It is often referred to simply as the elastic modulus.
The shear modulus or modulus of rigidity (G) describes an object's tendency to shear (the deformation of shape at constant volume) when acted upon by opposing forces; it is defined as shear stress over shear strain. The shear modulus is part of the derivation of viscosity.
The bulk modulus (K) describes volumetric elasticity, or the tendency of an object's volume to deform when under pressure; it is defined as volumetric stress over volumetric strain, and is the inverse of compressibility. The bulk modulus is an extension of Young's modulus to three dimensions.

[edit] Relationships between elastic moduli for isotropic media

For an isotropic elastic material the elastic moduli are related as follows: ( $ν$ is Poisson's ratio)

	$E\,$	$K\,$	$G\,$
$E=\,$		$3(1-2\nu)K\,$	$2G(1+\nu)\,$
$K=\,$	$\frac{E}{3(1-2\nu)}$		$\frac{2(1+\nu)G}{3(1-2\nu)}$
$G=\,$	$\frac{E}{2(1+\nu)}$	$\frac{3(1-2\nu)K}{2(1+\nu)}$