弹性模量

弹性模量（elastic modulus，modulus of elasticity，MOE）是理想材料在小形变时应力（低于材料比例极限的应力）与相应的应变之比值^[1]。

弹性模量指当有力施加於物体或物质时，其弹性变形（非永久变形）趋势的数学描述，为物体变形难易程度的表征，也是弹性材料最重要、最具特征的一种力学性质。

物体的弹性模量数学定义为：弹性变形区的应力－应变曲线的斜率：

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

其中λ是弹性模量，stress（应力）是引起受力区变形的力，strain（应变）是应力引起的变化与物体原始状态的比。应力的单位是帕斯卡，应变是没有单位的（无量纲的），那么λ的单位也是帕斯卡。

均质各向同性（固体）材料的（线性）弹性性质可以由4种弹性模量中的任意2种弹性模量完全描述清楚，如下表所示。

无粘性流体不能支撑剪切应力，因此剪切模量总为零，从而杨氏模量也总为零。

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。

• 刚度
• 弹性极限
• 弹性 (物理学)
• 杨氏模量
• 脉冲激励法
• 抗拉强度
• 彈性波
• 动模量
• 抗弯刚度
• 横观各向同性
• 弯曲模量

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

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。
	${\displaystyle (\lambda ,\,G)}$	${\displaystyle (E,\,G)}$	${\displaystyle (K,\,\lambda )}$	${\displaystyle (K,\,G)}$	${\displaystyle (\lambda ,\,\nu )}$	${\displaystyle (G,\,\nu )}$	${\displaystyle (E,\,\nu )}$	${\displaystyle (K,\,\nu )}$	${\displaystyle (K,\,E)}$	${\displaystyle (M,\,G)}$
${\displaystyle K=\,}$	${\displaystyle \lambda +{\tfrac {2G}{3}}}$	${\displaystyle {\tfrac {EG}{3(3G-E)}}}$			${\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}$	${\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}$	${\displaystyle {\tfrac {E}{3(1-2\nu )}}}$			${\displaystyle M-{\tfrac {4G}{3}}}$
${\displaystyle E=\,}$	${\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}$		${\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}$	${\displaystyle {\tfrac {9KG}{3K+G}}}$	${\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}$	${\displaystyle 2G(1+\nu )\,}$		${\displaystyle 3K(1-2\nu )\,}$		${\displaystyle {\tfrac {G(3M-4G)}{M-G}}}$
${\displaystyle \lambda =\,}$		${\displaystyle {\tfrac {G(E-2G)}{3G-E}}}$		${\displaystyle K-{\tfrac {2G}{3}}}$		${\displaystyle {\tfrac {2G\nu }{1-2\nu }}}$	${\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}$	${\displaystyle {\tfrac {3K\nu }{1+\nu }}}$	${\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}$	${\displaystyle M-2G\,}$
${\displaystyle G=\,}$			${\displaystyle {\tfrac {3(K-\lambda )}{2}}}$		${\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}$		${\displaystyle {\tfrac {E}{2(1+\nu )}}}$	${\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}$	${\displaystyle {\tfrac {3KE}{9K-E}}}$
${\displaystyle \nu =\,}$	${\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}$	${\displaystyle {\tfrac {E}{2G}}-1}$	${\displaystyle {\tfrac {\lambda }{3K-\lambda }}}$	${\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}$					${\displaystyle {\tfrac {3K-E}{6K}}}$	${\displaystyle {\tfrac {M-2G}{2M-2G}}}$
${\displaystyle M=\,}$	${\displaystyle \lambda +2G\,}$	${\displaystyle {\tfrac {G(4G-E)}{3G-E}}}$	${\displaystyle 3K-2\lambda \,}$	${\displaystyle K+{\tfrac {4G}{3}}}$	${\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}$	${\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}$	${\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}$	${\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}$	${\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}$