剪切模量

剪力模數
常見符號	G, S
国际单位	帕斯卡
從其他物理量的推衍	G = τ / γ
因次	${\displaystyle {\mathsf {L}}^{-1}{\mathsf {M}}{\mathsf {T}}^{-2}}$

剪力模數（shear modulus）是材料力學中的名詞，彈性材料承受剪應力時會產生剪應變，定義為剪應力與剪應變的比值。公式記為

${\displaystyle G={\frac {\tau }{\gamma }}}$

其中，${\displaystyle G\,}$ 表示剪力模數，${\displaystyle \tau \,}$ 表示剪應力，${\displaystyle \gamma \,}$ 表示剪應變。在均質且等向性的材料中：

${\displaystyle G={E \over {2(1+\nu )}}}$

其中，${\displaystyle E\,}$ 是楊氏模數(Young's modulus )，${\displaystyle \nu \,}$ 是泊松比(Poisson's ratio)。

在均匀各向同性固体中，有两种波:P波和S波。剪切波的速度，${\displaystyle (v_{s})}$由剪切模量控制，

${\displaystyle v_{s}={\sqrt {\frac {G}{\rho }}}}$

其中

G是剪切模量

${\displaystyle \rho }$是固体的密度.

金属的剪切模量通常随温度的升高而降低。在高压下，剪切模量也随外加压力的增大而增大。在许多金属中，熔点温度、空位形成能和剪切模量之间的关系已经被观察到。^[3]

有几种模型试图预测金属的剪切模量(可能还有合金的剪切模量)。在塑性流动计算中使用的剪切模量模型包括:

1. MTS剪切模量模型由机械阈值应力(MTS)塑性流动应力模型开发并与之结合使用。^[4][5][6]
2. 由SCGL流动应力模型开发并与之结合使用的SCGL剪切模量模型。^[7]
3. 纳达尔和LePoac (NP)剪切模量模型，利用Lindemann理论确定剪切模量对温度的依赖关系，利用SCG模型确定剪切模量对压力的依赖关系。^[2]

MTS剪切模量模型为:

${\displaystyle \mu (T)=\mu _{0}-{\frac {D}{\exp(T_{0}/T)-1}}}$

其中${\displaystyle \mu _{0}}$为${\displaystyle T=0K}$处的剪切模量，${\displaystyle D}$和${\displaystyle T_{0}}$为材料常数。

• 固體力學
• 流體力學
• 連續介質力學

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

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2. ^ ^{2.0 2.1} Nadal, Marie-Hélène; Le Poac, Philippe. Continuous model for the shear modulus as a function of pressure and temperature up to the melting point: Analysis and ultrasonic validation. Journal of Applied Physics. 2003, 93 (5): 2472. Bibcode:2003JAP....93.2472N. doi:10.1063/1.1539913. 
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