拉梅常數

在線性彈性力學中，拉梅常數包括以下兩個常數：

• 拉梅常數λ，又稱拉梅第一常數
• 剪切模數μ，又稱拉梅第二常數，也可記為${\displaystyle G}$

上述常數的條件是等向性材料，並在三維中滿足虎克定律。

${\displaystyle \sigma =2\mu \varepsilon +\lambda \;\mathrm {tr} (\varepsilon )I}$

其中σ是應力，ε是體積變形量，${\displaystyle \scriptstyle I}$是單位矩陣，${\displaystyle \scriptstyle \mathrm {tr} (\cdot )}$是跡數函數。

第一常數λ沒有物理解釋，但其有助於化簡胡克定律的剛度矩陣。兩個常數構建了均質等向性介質的彈性模數的常數化形式，並與其他彈性模數形成了聯繫。

拉梅常數以拉梅（Gabriel Lamé）的名字命名。

參考文獻

• F. Kang, S. Zhong-Ci, Mathematical Theory of Elastic Structures, Springer New York, ISBN 0-387-51326-4, (1981)
• G. Mavko, T. Mukerji, J. Dvorkin, The Rock Physics Handbook, Cambridge University Press (paperback), ISBN 0-521-54344-4, (2003)

換算公式
均質等向性線彈性材料具有獨特的彈性性質，因此知道彈性模數中的任意兩種，就可由下列換算公式求出其他所有的彈性模數。
	${\displaystyle (\lambda ,\,G)}$	${\displaystyle (E,\,G)}$	${\displaystyle (K,\,\lambda )}$	${\displaystyle (K,\,G)}$	${\displaystyle (\lambda ,\,\nu )}$	${\displaystyle (G,\,\nu )}$	${\displaystyle (E,\,\nu )}$	${\displaystyle (K,\,\nu )}$	${\displaystyle (K,\,E)}$	${\displaystyle (M,\,G)}$
${\displaystyle K=\,}$	${\displaystyle \lambda +{\tfrac {2G}{3}}}$	${\displaystyle {\tfrac {EG}{3(3G-E)}}}$			${\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}$	${\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}$	${\displaystyle {\tfrac {E}{3(1-2\nu )}}}$			${\displaystyle M-{\tfrac {4G}{3}}}$
${\displaystyle E=\,}$	${\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}$		${\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}$	${\displaystyle {\tfrac {9KG}{3K+G}}}$	${\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}$	${\displaystyle 2G(1+\nu )\,}$		${\displaystyle 3K(1-2\nu )\,}$		${\displaystyle {\tfrac {G(3M-4G)}{M-G}}}$
${\displaystyle \lambda =\,}$		${\displaystyle {\tfrac {G(E-2G)}{3G-E}}}$		${\displaystyle K-{\tfrac {2G}{3}}}$		${\displaystyle {\tfrac {2G\nu }{1-2\nu }}}$	${\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}$	${\displaystyle {\tfrac {3K\nu }{1+\nu }}}$	${\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}$	${\displaystyle M-2G\,}$
${\displaystyle G=\,}$			${\displaystyle {\tfrac {3(K-\lambda )}{2}}}$		${\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}$		${\displaystyle {\tfrac {E}{2(1+\nu )}}}$	${\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}$	${\displaystyle {\tfrac {3KE}{9K-E}}}$
${\displaystyle \nu =\,}$	${\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}$	${\displaystyle {\tfrac {E}{2G}}-1}$	${\displaystyle {\tfrac {\lambda }{3K-\lambda }}}$	${\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}$					${\displaystyle {\tfrac {3K-E}{6K}}}$	${\displaystyle {\tfrac {M-2G}{2M-2G}}}$
${\displaystyle M=\,}$	${\displaystyle \lambda +2G\,}$	${\displaystyle {\tfrac {G(4G-E)}{3G-E}}}$	${\displaystyle 3K-2\lambda \,}$	${\displaystyle K+{\tfrac {4G}{3}}}$	${\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}$	${\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}$	${\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}$	${\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}$	${\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}$