Longitudinal module

The Longitudinalmodul , also longitudinal module or P-wave module called, is one of elastic moduli , in the linear theory of elasticity for describing homogeneous isotropic can be used materials. ${\displaystyle M}$

In a uniaxial stress state in which all non-axial deformations are equal to ( with ) zero, the longitudinal modulus is defined as the ratio of axial stress to axial deformation : ${\displaystyle \epsilon _{**}=0}$ ${\displaystyle *\neq z}$ ${\displaystyle \sigma _{zz}}$ ${\displaystyle \epsilon _{zz}}$

${\displaystyle {\begin{aligned}M&={\frac {\sigma _{zz}}{\epsilon _{zz}}}\\\Leftrightarrow \sigma _{zz}&=M\,\epsilon _{zz}\end{aligned}}}$

As I said, this only applies if all components of the elasticity tensor except for one diagonal component are zero. Because of its property as a diagonal component of the elasticity tensor, the longitudinal module is sometimes also referred to with the symbol . ${\displaystyle C_{11}}$

The following applies to the longitudinal module:

${\displaystyle M=\rho \,v_{p}^{2}}$

With

• the density of the material ${\displaystyle \rho }$
• the speed of the longitudinal waves or P waves . ${\displaystyle v_{p}}$

Therefore, the longitudinal modulus can be determined by measuring the velocity of ultrasound in a material.

Also applies

${\displaystyle M={\frac {1-\nu }{1-\nu -2\nu ^{2}}}\,E}$

With

• the modulus of elasticity ${\displaystyle E}$
• the Poisson's number . ${\displaystyle \nu }$

If the transverse contraction ( ) is negligible . ${\displaystyle \nu \approx 0}$ ${\displaystyle M\approx E}$

See also

• Conversion between the elastic constants

literature

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

Individual evidence

1. ↑ Thomas Lüthi: Contribution to the determination of residual stresses with ultrasound. Application on weld seams. Report No. 219 . Eidgenössische Materialprüfungs- und Forschungsanstalt, Dübendorf 1990. Archived from the original on November 1, 2012 Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (Accessed February 4, 2016). 
2. ↑ HM Ledbetter: Elastic Constants of Polycrystalline Copper at Low Temperatures . In: physica status solidi (a) . 66, 1981, pp. 477-484. Retrieved February 4, 2016.