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# Elastic modulus

An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be 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:

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

where λ (lambda) 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. If stress is measured in pascals, and since strain is a unitless ratio, then the units of λ are 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.

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

• 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 or μ) 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.

Three other elastic moduli are Poisson's ratio, 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.

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

v  d  e Elastic moduli for homogeneous isotropic materials

Bulk modulus (K) | Young's modulus (E) | Lamé's first parameter (λ) | Shear modulus (μ) | Poisson's ratio (ν) | P-wave modulus (M)

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.
$(\lambda,\,\mu)$ $(E,\,\mu)$ $(K,\,\lambda)$ $(K,\,\mu)$ $(\lambda,\,\nu)$ $(\mu,\,\nu)$ $(E,\,\nu)$ $(K,\, \nu)$ $(K,\,E)$
$K=\,$ $\lambda+ \frac{2\mu}{3}$ $\frac{E\mu}{3(3\mu-E)}$ $\lambda\frac{1+\nu}{3\nu}$ $\frac{2\mu(1+\nu)}{3(1-2\nu)}$ $\frac{E}{3(1-2\nu)}$
$E=\,$ $\mu\frac{3\lambda + 2\mu}{\lambda + \mu}$ $9K\frac{K-\lambda}{3K-\lambda}$ $\frac{9K\mu}{3K+\mu}$ $\frac{\lambda(1+\nu)(1-2\nu)}{\nu}$ $2\mu(1+\nu)\,$ $3K(1-2\nu)\,$
$\lambda=\,$ $\mu\frac{E-2\mu}{3\mu-E}$ $K-\frac{2\mu}{3}$ $\frac{2 \mu \nu}{1-2\nu}$ $\frac{E\nu}{(1+\nu)(1-2\nu)}$ $\frac{3K\nu}{1+\nu}$ $\frac{3K(3K-E)}{9K-E}$
$\mu=\,$ $3\frac{K-\lambda}{2}$ $\lambda\frac{1-2\nu}{2\nu}$ $\frac{E}{2+2\nu}$ $3K\frac{1-2\nu}{2+2\nu}$ $\frac{3KE}{9K-E}$
$\nu=\,$ $\frac{\lambda}{2(\lambda + \mu)}$ $\frac{E}{2\mu}-1$ $\frac{\lambda}{3K-\lambda}$ $\frac{3K-2\mu}{2(3K+\mu)}$ $\frac{3K-E}{6K}$
$M=\,$ $\lambda+2\mu\,$ $\mu\frac{4\mu-E}{3\mu-E}$ $3K-2\lambda\,$ $K+\frac{4\mu}{3}$ $\lambda \frac{1-\nu}{\nu}$ $\mu\frac{2-2\nu}{1-2\nu}$ $E\frac{1-\nu}{(1+\nu)(1-2\nu)}$ $3K\frac{1-\nu}{1+\nu}$ $3K\frac{3K+E}{9K-E}$