P波模量

在线性弹性力学中，P波模量$M$，或称纵向弹性模量是用于描述各向同性均质材料的弹性模量。

其定义为单轴应变下轴向应力与轴向应变的比：

$\sigma _{zz}=M\epsilon _{zz}$

而其他所有的应变$\epsilon _{**}$均为零。

上式等价于

$M=\rho V_{p}^{2}$

其中$V_{P}$是P波速度。

参考文献

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

换算公式
均质各向同性线弹性材料具有独特的弹性性质，因此知道弹性模量中的任意两种，就可由下列换算公式求出其他所有的弹性模量。
	$(\lambda ,\,G)$	$(E,\,G)$	$(K,\,\lambda )$	$(K,\,G)$	$(\lambda ,\,\nu )$	$(G,\,\nu )$	$(E,\,\nu )$	$(K,\,\nu )$	$(K,\,E)$	$(M,\,G)$
$K=\,$	$\lambda +{\tfrac {2G}{3}}$	${\tfrac {EG}{3(3G-E)}}$			${\tfrac {\lambda (1+\nu )}{3\nu }}$	${\tfrac {2G(1+\nu )}{3(1-2\nu )}}$	${\tfrac {E}{3(1-2\nu )}}$			$M-{\tfrac {4G}{3}}$
$E=\,$	${\tfrac {G(3\lambda +2G)}{\lambda +G}}$		${\tfrac {9K(K-\lambda )}{3K-\lambda }}$	${\tfrac {9KG}{3K+G}}$	${\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}$	$2G(1+\nu )\,$		$3K(1-2\nu )\,$		${\tfrac {G(3M-4G)}{M-G}}$
$\lambda =\,$		${\tfrac {G(E-2G)}{3G-E}}$		$K-{\tfrac {2G}{3}}$		${\tfrac {2G\nu }{1-2\nu }}$	${\tfrac {E\nu }{(1+\nu )(1-2\nu )}}$	${\tfrac {3K\nu }{1+\nu }}$	${\tfrac {3K(3K-E)}{9K-E}}$	$M-2G\,$
$G=\,$			${\tfrac {3(K-\lambda )}{2}}$		${\tfrac {\lambda (1-2\nu )}{2\nu }}$		${\tfrac {E}{2(1+\nu )}}$	${\tfrac {3K(1-2\nu )}{2(1+\nu )}}$	${\tfrac {3KE}{9K-E}}$
$\nu =\,$	${\tfrac {\lambda }{2(\lambda +G)}}$	${\tfrac {E}{2G}}-1$	${\tfrac {\lambda }{3K-\lambda }}$	${\tfrac {3K-2G}{2(3K+G)}}$					${\tfrac {3K-E}{6K}}$	${\tfrac {M-2G}{2M-2G}}$
$M=\,$	$\lambda +2G\,$	${\tfrac {G(4G-E)}{3G-E}}$	$3K-2\lambda \,$	$K+{\tfrac {4G}{3}}$	${\tfrac {\lambda (1-\nu )}{\nu }}$	${\tfrac {2G(1-\nu )}{1-2\nu }}$	${\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}$	${\tfrac {3K(1-\nu )}{1+\nu }}$	${\tfrac {3K(3K+E)}{9K-E}}$