Elastic modulus

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Elastic modulus is a generic name for a set of parameters that describe the elastic properties of materials:

The name Elastic modulus is sometimes used to refer to Young's modulus. This is rather ambiguous and best avoided.

The table shown below first appeared, I think, in Gassmann[1]


  1. Gassmann, F (1951). Über die Elastizität poröser Medien. Viertel. Naturforsch. Ges. Zürich 96, 1–23. [ageo.co/gassmann-de Available online.]

Table of equations

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} \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} \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
\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
\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
\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
\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
\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
\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
\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}