# Compressibility

Compressibility is the normalized change of density with respect to pressure, or equivalently, the negative of the change of specific volume with pressure. It is the inverse of the incompressibility.

Hooke's law for the specific volume change is:

 ${\displaystyle {\frac {\Delta V}{V}}=\sum _{i}^{3}\varepsilon _{ii}=\sum _{i,k,l}^{3}S_{iikl}\Delta \sigma _{kl}}$ (1)

where ${\displaystyle \varepsilon }$ is strain and ${\displaystyle \Delta \sigma }$ is the increment of stress. ${\displaystyle {\textbf {S}}}$ is the rank-4 compliance tensor. When the stress is a pure pressure,

 ${\displaystyle \sigma _{kl}={\begin{pmatrix}-\Delta P&0&0\\0&-\Delta P&0\\0&0&-\Delta P\\\end{pmatrix}}=-\Delta P\delta _{kl}}$ (2)

with ${\displaystyle \delta }$ the Kronecker delta (the rank-2 identity tensor). The minus sign is the standard convention relating stress and pressure. Using equation (2) in equation (1),

 ${\displaystyle {\frac {\Delta V}{V}}=-\Delta P\sum _{i,k}^{3}S_{iikk}}$ (3)

Switching to the 2-index (Voigt notation) for the compliance matrix, the compressibility ${\displaystyle \kappa =}$ is given by:

 ${\displaystyle \kappa ={\frac {-1}{V}}{\frac {\Delta V}{\Delta P}}=\sum _{\alpha ,\beta }^{3}S_{\alpha \beta }}$ (4)

For isotropic media, this yields the standard relationship between ${\displaystyle \kappa }$, Young's modulus ${\displaystyle E}$ and Poisson's ratio ${\displaystyle \nu }$. (For such isotropic media, it is also true that

 ${\displaystyle {\frac {1}{\kappa ^{iso}}}={\frac {1}{9}}\sum _{I,K}^{3}C_{IK}}$ (5)

but this is not true in general.) For anisotropic media, equation (4) gives the scalar change in specific volume, although the shape of the medium may change.

This development is not concerned with the thermodynamics of the deformation. But, one usually assumes that, for wave propagation, the deformation is adiabatic, and for geomechanics, the deformation may be isothermal; the compressibility components are slightly different in these two cases.