彈性模量

彈性模量是指當有力施加於物體或物質時，其彈性變形（非永久變形）趨勢的數學描述。物體的彈性模量定義為彈性變形區的應力－應變曲線的斜率：

${\displaystyle \lambda \ {\stackrel {\text{def}}{=}}\ {\frac {\text{stress}}{\text{strain}}}}$

其中λ是彈性模量，stress（應力）是引起受力區變形的力，strain（應變）是應力引起的變化與物體原始狀態的比。應力的單位是帕斯卡，應變是沒有單位的（無量綱的），那麼λ的單位也是帕斯卡。

均質各向同性（固體）材料的（線性）彈性性質可以由4種彈性模量中的任意2種彈性模量完全描述清楚，如下表所示。

無粘性流體不能支撐剪切應力，因此剪切模量總為零，從而楊氏模量也總為零。

檢測方法

彈性模量檢測方法分為靜態法（例如靜荷重法）和動態法（例如共振法）兩種。使用動態法產生的結果就是動模量或稱動彈性模量。

參見

• 剛度
• 彈性極限
• 彈性 (物理學)
• 楊氏模量
• 脈衝激勵法
• 抗拉強度
• 彈性波
• 動模量
• 抗彎剛度
• 橫觀各向同性
• 彎曲模量

參考文獻

• C. Hartsuijker, J.W. Welleman. Engineering Mechanics Volume 2. Springer. 2001. ISBN 978-1-4020-4123-5.

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