مدول برشی

در دانش مواد، مدول برشی یا مدول صلبیت که با G نمایش داده می‌شود که برابر است با نسبت تنش برشی به کرنش برشی:[1]

${\displaystyle G\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\tau _{xy}}{\gamma _{xy}}}={\frac {F/A}{\Delta x/l}}={\frac {Fl}{A\Delta x}}}$

کرنش برشی

که در آن

${\displaystyle \tau _{xy}=F/A\,}$ تنش برشی.

${\displaystyle F}$ نیروی وارده.

${\displaystyle A}$ سطحی که نیرو بر آن اثر می‌کند.

در مهندسی ${\displaystyle \gamma _{xy}=\Delta x/l=\tan \theta \,}$ و در جاهای دیگر ${\displaystyle \gamma _{xy}=\theta }$.

${\displaystyle \Delta x}$ جابجایی عرضی است.

${\displaystyle l}$ طول اولیه‌است.

علاوه بر G گاهی مدول برشی را با S یا μ نیز نمایش می‌دهند. یکای مدول برشی گیگاپاسکال (GPa) یا هزار پوند بر اینچ مربع (ksi) است. مدول برشی همواره مثبت است.

یادداشت و منبع

• ویکی‌پدیای انگلیسی

1. IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "shear modulus, G".

رابطه‌های تبدیل مدول‌ها به یکدیگر
خواص کشسانی مواد کشسان خطی همگن و همسانگرد را می‌توان با داشتن دو مدول دلخواه به طور کامل و منحصر به فردی تعیین کرد. بنابراین با در دست داشتن دو مدول و با استفاده از فرمول‌های زیر می‌توان سایر مدول‌ها را محاسبه کرد.
	${\displaystyle K=\,}$	${\displaystyle E=\,}$	${\displaystyle \lambda =\,}$	${\displaystyle G=\,}$	${\displaystyle \nu =\,}$	${\displaystyle M=\,}$	توضیحات
${\displaystyle (K,\,E)}$	${\displaystyle K}$	${\displaystyle E}$	${\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}$	${\displaystyle {\tfrac {3KE}{9K-E}}}$	${\displaystyle {\tfrac {3K-E}{6K}}}$	${\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}$
${\displaystyle (K,\,\lambda )}$	${\displaystyle K}$	${\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}$	${\displaystyle \lambda }$	${\displaystyle {\tfrac {3(K-\lambda )}{2}}}$	${\displaystyle {\tfrac {\lambda }{3K-\lambda }}}$	${\displaystyle 3K-2\lambda \,}$
${\displaystyle (K,\,G)}$	${\displaystyle K}$	${\displaystyle {\tfrac {9KG}{3K+G}}}$	${\displaystyle K-{\tfrac {2G}{3}}}$	${\displaystyle G}$	${\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}$	${\displaystyle K+{\tfrac {4G}{3}}}$
${\displaystyle (K,\,\nu )}$	${\displaystyle K}$	${\displaystyle 3K(1-2\nu )\,}$	${\displaystyle {\tfrac {3K\nu }{1+\nu }}}$	${\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}$	${\displaystyle \nu }$	${\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}$
${\displaystyle (K,\,M)}$	${\displaystyle K}$	${\displaystyle {\tfrac {9K(M-K)}{3K+M}}}$	${\displaystyle {\tfrac {3K-M}{2}}}$	${\displaystyle {\tfrac {3(M-K)}{4}}}$	${\displaystyle {\tfrac {3K-M}{3K+M}}}$	${\displaystyle M}$
${\displaystyle (E,\,\lambda )}$	${\displaystyle {\tfrac {E+3\lambda +R}{6}}}$	${\displaystyle E}$	${\displaystyle \lambda }$	${\displaystyle {\tfrac {E-3\lambda +R}{4}}}$	${\displaystyle {\tfrac {2\lambda }{E+\lambda +R}}}$	${\displaystyle {\tfrac {E-\lambda +R}{2}}}$	${\displaystyle R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}$
${\displaystyle (E,\,G)}$	${\displaystyle {\tfrac {EG}{3(3G-E)}}}$	${\displaystyle E}$	${\displaystyle {\tfrac {G(E-2G)}{3G-E}}}$	${\displaystyle G}$	${\displaystyle {\tfrac {E}{2G}}-1}$	${\displaystyle {\tfrac {G(4G-E)}{3G-E}}}$
${\displaystyle (E,\,\nu )}$	${\displaystyle {\tfrac {E}{3(1-2\nu )}}}$	${\displaystyle E}$	${\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}$	${\displaystyle {\tfrac {E}{2(1+\nu )}}}$	${\displaystyle \nu }$	${\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}$
${\displaystyle (E,\,M)}$	${\displaystyle {\tfrac {3M-E+S}{6}}}$	${\displaystyle E}$	${\displaystyle {\tfrac {M-E+S}{4}}}$	${\displaystyle {\tfrac {3M+E-S}{8}}}$	${\displaystyle {\tfrac {E-M+S}{4M}}}$	${\displaystyle M}$	${\displaystyle S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}}$ There are two valid solutions. The plus sign leads to ${\displaystyle \nu \geq 0}$. The minus sign leads to ${\displaystyle \nu \leq 0}$.
${\displaystyle (\lambda ,\,G)}$	${\displaystyle \lambda +{\tfrac {2G}{3}}}$	${\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}$	${\displaystyle \lambda }$	${\displaystyle G}$	${\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}$	${\displaystyle \lambda +2G\,}$
${\displaystyle (\lambda ,\,\nu )}$	${\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}$	${\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}$	${\displaystyle \lambda }$	${\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}$	${\displaystyle \nu }$	${\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}$	Cannot be used when ${\displaystyle \nu =0\Leftrightarrow \lambda =0}$
${\displaystyle (\lambda ,\,M)}$	${\displaystyle {\tfrac {M+2\lambda }{3}}}$	${\displaystyle {\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}}$	${\displaystyle \lambda }$	${\displaystyle {\tfrac {M-\lambda }{2}}}$	${\displaystyle {\tfrac {\lambda }{M+\lambda }}}$	${\displaystyle M}$
${\displaystyle (G,\,\nu )}$	${\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}$	${\displaystyle 2G(1+\nu )\,}$	${\displaystyle {\tfrac {2G\nu }{1-2\nu }}}$	${\displaystyle G}$	${\displaystyle \nu }$	${\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}$
${\displaystyle (G,\,M)}$	${\displaystyle M-{\tfrac {4G}{3}}}$	${\displaystyle {\tfrac {G(3M-4G)}{M-G}}}$	${\displaystyle M-2G\,}$	${\displaystyle G}$	${\displaystyle {\tfrac {M-2G}{2M-2G}}}$	${\displaystyle M}$
${\displaystyle (\nu ,\,M)}$	${\displaystyle {\tfrac {M(1+\nu )}{3(1-\nu )}}}$	${\displaystyle {\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}}$	${\displaystyle {\tfrac {M\nu }{1-\nu }}}$	${\displaystyle {\tfrac {M(1-2\nu )}{2(1-\nu )}}}$	${\displaystyle \nu }$	${\displaystyle M}$