Young's modulus

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Definition

Young's Modulus, or lambda E, is an elastic modulus is a measure of the stiffness of a material. It is used extensively in quantitative seismic interpretation, rock physics, and rock mechanics. It is defined as the ratio of uniaxial stress to uniaxial strain when linear elasticity applies. It is analogous to the proportionality constant in Hooke's Law F=-kx.

Definition

\textrm{Young's\ modulus} = \frac{\textrm{longitudinal\ stress}}{\textrm{longitudinal\ strain}}
E = \frac{F / W^2}{\Delta L / L}

Where W 2 is the cross-sectional area of the material (see figure).

In terms of VP and VS

E=\frac{\rho V_\mathrm{S}^2\left ( 3V_\mathrm{P}^2-4V_\mathrm{S}^2 \right )}{V_\mathrm{P}^2-V_\mathrm{S}^2}

Here's a function you can paste into Excel to compute Young's modulus (see Using equations in Excel):

 =rho*VsS-wave velocity^2*(3*VpP-wave velocity^2-4*VsS-wave velocity^2)/(VpP-wave velocity^2-VsS-wave velocity^2)

And here is the same code for definition as a function in VBA:

Public Function Youngs(VpP-wave velocity As Double, VsS-wave velocity As Double, rho As Double) As Double
  Youngs = rho * VsS-wave velocity ^ 2 * (3 * VpP-wave velocity ^ 2 - 4 * VsS-wave velocity ^ 2) / (VpP-wave velocity ^ 2 - VsS-wave velocity ^ 2)
End Function

Other expressions

E can also be expressed in terms of the first Lamé parameter λ and shear modulus μ:

E = \frac{\mu(3\lambda+2\mu)}{\lambda+\mu}

The simplest expression casts E in terms of bulk modulus K and shear modulus μ:

E = \frac{9K\mu}{3K+\mu}

The (E, λ) problem

The other elastic moduli are not pretty when expressed in terms of E and λ:

\mu = \frac14 \left( -3\lambda + E \pm \sqrt{9\lambda^2 + 2\lambda E + E^2} \right)
 K = \frac16 \left( 3\lambda + E \pm \sqrt{9\lambda^2 + 2\lambda E + E^2} \right)
\nu = \frac{1}{4\lambda} \left( -\lambda - E \pm \sqrt{9\lambda^2 + 2\lambda E + E^2} \right)
 M = \frac12 \left( -\lambda + E \pm \sqrt{9\lambda^2 + 2\lambda E + E^2} \right)

What is this quantity \sqrt{9\lambda^2 + 2\lambda E + E^2} ?

References

External links

find literature about
Young's modulus
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Conversion formulas — edit
The elastic properties of homogeneous isotropic linear elastic materials are uniquely determined by any two moduli. Given any two, the others can thus be calculated. Key reference: Mavko, G, T Mukerji and J Dvorkin (2003), The Rock Physics Handbook, Cambridge University Press.

\dagger \ X = \sqrt{9\lambda^2 + 2E\lambda + E^2}

(V_\mathrm{P},\,V_\mathrm{S}) (\mu,\,\lambda) (E,\,\lambda)\,\dagger (E,\,\mu) (K,\,\lambda) (K,\,\mu) (K,\,E) (\nu,\,\lambda) (\nu,\,\mu) (\nu,\,E) (\nu,\,K)
P-wave velocity
V_\mathrm{P}=\,
V_\mathrm{P} \sqrt{\tfrac{\lambda+2\mu}{\rho}} \sqrt{\tfrac{E - \lambda + X}{2\rho}} \sqrt{\tfrac{\mu\,(E-4\mu)}{\rho\,(E-3\mu)}} \sqrt{\tfrac{3K - 2\lambda}{\rho}} \sqrt{\tfrac{K+\frac43\mu}{\rho}} \sqrt{\tfrac{3K \left(3K + E\right)}{\rho \left(9K - E\right)}} \sqrt{\tfrac{\lambda (1 - \nu)}{\rho\nu}} \sqrt{\tfrac{2 \mu (1-\nu)}{\rho (1 - 2\nu)}} \sqrt{\tfrac{E(1 - \nu)}{\rho(1 + \nu)(1 - 2\nu)}} \sqrt{\tfrac{3K(1 - \nu)}{\rho(1 + \nu)}}
S-wave velocity
V_\mathrm{S}=\,
V_\mathrm{S} \sqrt{\tfrac{\mu}{\rho}} \sqrt{\tfrac{E - 3\lambda + X}{4\rho}} \sqrt{\tfrac{\mu}{\rho}} \sqrt{\tfrac{3 (K - \lambda)}{2 \rho}} \sqrt{\tfrac{\mu}{\rho}} \sqrt{- \tfrac{3EK}{\rho \left(E - 9K\right)}} \sqrt{\tfrac{\lambda}{2 \nu \rho} - \tfrac{\lambda}{\rho}} \sqrt{\tfrac{\mu}{\rho}} \sqrt{\tfrac{E}{2\rho(1+\nu)}} \sqrt{-\tfrac{3K (2\nu - 1)}{2\rho (\nu + 1)}}
Velocity ratio
\Gamma=\,
\frac{V_\mathrm{P}}{V_\mathrm{S}} \sqrt{\tfrac{\lambda+2\mu}{\mu}} \sqrt{\tfrac{3E + 3\lambda + X}{2E}} \sqrt{\tfrac{E - 4 \mu}{E - 3 \mu}} \sqrt{\tfrac{\tfrac43 \lambda - 2 K}{\lambda - K}} \sqrt{\tfrac{K+\frac43\mu}{\mu}} \sqrt{\tfrac{E + 3K}{E}} \sqrt{\tfrac{2\nu - 2}{2\nu - 1}} \sqrt{\tfrac{2\nu - 2}{2\nu - 1}} \sqrt{\tfrac{2\nu - 2}{2\nu - 1}} \sqrt{\tfrac{2\nu - 2}{2\nu - 1}}
1st Lamé parameter
\lambda=\,
\rho (V_\mathrm{P}^2 - 2V_\mathrm{S}^2) \lambda \lambda \tfrac{\mu(E-2\mu)}{3\mu-E} \lambda K-\tfrac{2\mu}{3} \tfrac{3K(3K-E)}{9K-E} \lambda \tfrac{2 \mu \nu}{1-2\nu} \tfrac{E\nu}{(1+\nu)(1-2\nu)} \tfrac{3K\nu}{1+\nu}
Shear modulus
\mu=\,
\rho V_\mathrm{S}^2 \mu \tfrac{E - 3\lambda + X}{4} \mu \tfrac{3(K-\lambda)}{2} \mu \tfrac{3KE}{9K-E} \tfrac{\lambda(1-2\nu)}{2\nu} \mu \tfrac{E}{2(1+\nu)} \tfrac{3K(1-2\nu)}{2(1+\nu)}
Young's modulus
E=\,
\tfrac{\rho V_\mathrm{S}^2 (3V_\mathrm{P}^2 - 4V_\mathrm{S}^2)}{V_\mathrm{P}^2 - V_\mathrm{S}^2} \tfrac{\mu(3\lambda + 2\mu)}{\lambda + \mu} E\ E \tfrac{9K(K-\lambda)}{3K-\lambda} \tfrac{9K\mu}{3K+\mu} E\ \tfrac{\lambda(1+\nu)(1-2\nu)}{\nu} 2\mu(1+\nu)\, E\ 3K(1-2\nu)\,
Bulk modulus
K=\,
\rho (V_\mathrm{P}^2 - \tfrac43 V_\mathrm{S}^2) \lambda+ \tfrac{2\mu}{3} \tfrac{E + 3\lambda + X}{6} \tfrac{E\mu}{3(3\mu-E)} K K K \tfrac{\lambda(1+\nu)}{3\nu} \tfrac{2\mu(1+\nu)}{3(1-2\nu)} \tfrac{E}{3(1-2\nu)} K
Poisson's ratio
\nu=\,
\tfrac{V_\mathrm{P}^2 - 2V_\mathrm{S}^2}{2(V_\mathrm{P}^2 - V_\mathrm{S}^2)} \tfrac{\lambda}{2(\lambda + \mu)} \tfrac{- E - \lambda + X}{4\lambda} \tfrac{E}{2\mu}-1 \tfrac{\lambda}{3K-\lambda} \tfrac{3K-2\mu}{2(3K+\mu)} \tfrac{3K-E}{6K} \nu \nu \nu \nu
P-wave modulus
M=\,
\rho V_\mathrm{P}^2 \lambda+2\mu\, \tfrac{E - \lambda + X}{2} \tfrac{\mu(4\mu-E)}{3\mu-E} 3K-2\lambda\, K+\tfrac{4\mu}{3} \tfrac{3K(3K+E)}{9K-E} \tfrac{\lambda(1-\nu)}{\nu} \tfrac{2\mu(1-\nu)}{1-2\nu} \tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)} \tfrac{3K(1-\nu)}{1+\nu}