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# P-wave modulus

In linear elasticity, the P-wave modulus M is one of the elastic moduli available to describe isotropic homogeneous materials.

It is defined as the ratio of axial stress to axial strain in a uniaxial strain state

σzz = Mεzz

where all the other strains ε * * are zero.

This is equivalent to stating that

$M = \rho V_p^2$

where VP is the velocity of a P-wave.

## References

• G. Mavko, T. Mukerji, J. Dvorkin. The Rock Physics Handbook. Cambridge University Press 2003 (paperback). ISBN 0-521-54344-4

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