अपरूपण गुणांक

पदार्थ विज्ञान में, अपरूपण प्रतिबल और अपरूपण विकृति के अनुपात को अपरूपण गुणांक (shear modulus' या modulus of rigidity) कहते हैं। इसे G (कभी-कभी S याr μ ) से निरूपित किया जाता है। :^[1]

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

जहाँ

${\displaystyle \tau _{xy}=F/A\,}$ = अपरूपण प्रतिबल (shear stress);

${\displaystyle F}$ लगने वाला बल

${\displaystyle A}$ क्षेत्रफल, जिस पर बल लग रहा है

इंजीनियरी में , ${\displaystyle \gamma _{xy}=\Delta x/l=\tan \theta \,}$ = अपरूपण विकृति (shear strain). अन्य स्थानों पर , ${\displaystyle \gamma _{xy}=\theta }$

${\displaystyle \Delta x}$ अनुप्रस्थ विस्थापन

${\displaystyle l}$ मूल लम्बाई

अपरूपण गुणांक की एस आई इकाई पास्कल (Pa) है। इसकी विमा (dimension) M¹L⁻¹T⁻² है।

अपरूपण गुणांक सदा धनात्मक होता है।

परिवर्तन के सूत्र
होमोजिनस आइसोट्रॉपिक रैखिक प्रत्यास्थ पदार्थ के कोई भी दो मापांक दिये हों तो अन्य गुण निम्नलिखित सूत्रों द्वारा प्राप्त किये जा सकते हैं।
	${\displaystyle (K,\,E)}$	${\displaystyle (K,\,\lambda )}$	${\displaystyle (K,\,G)}$	${\displaystyle (K,\,\nu )}$	${\displaystyle (E,\,G)}$	${\displaystyle (E,\,\nu )}$	${\displaystyle (\lambda ,\,G)}$	${\displaystyle (\lambda ,\,\nu )}$	${\displaystyle (G,\,\nu )}$	${\displaystyle (G,\,M)}$
${\displaystyle K=\,}$	${\displaystyle K}$	${\displaystyle K}$	${\displaystyle K}$	${\displaystyle K}$	${\displaystyle {\tfrac {EG}{3(3G-E)}$	${\displaystyle {\tfrac {E}{3(1-2\nu )}$	${\displaystyle \lambda +{\tfrac {2G}{3}$	${\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }$	${\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}$	${\displaystyle M-{\tfrac {4G}{3}$
${\displaystyle E=\,}$	${\displaystyle E}$	${\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }$	${\displaystyle {\tfrac {9KG}{3K+G}$	${\displaystyle 3K(1-2\nu )\,}$	${\displaystyle E}$	${\displaystyle E}$	${\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}$	${\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }$	${\displaystyle 2G(1+\nu )\,}$	${\displaystyle {\tfrac {G(3M-4G)}{M-G}$
${\displaystyle \lambda =\,}$	${\displaystyle {\tfrac {3K(3K-E)}{9K-E}$	${\displaystyle \lambda }$	${\displaystyle K-{\tfrac {2G}{3}$	${\displaystyle {\tfrac {3K\nu }{1+\nu }$	${\displaystyle {\tfrac {G(E-2G)}{3G-E}$	${\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}$	${\displaystyle \lambda }$	${\displaystyle \lambda }$	${\displaystyle {\tfrac {2G\nu }{1-2\nu }$	${\displaystyle M-2G\,}$
${\displaystyle G=\,}$	${\displaystyle {\tfrac {3KE}{9K-E}$	${\displaystyle {\tfrac {3(K-\lambda )}{2}$	${\displaystyle G}$	${\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}$	${\displaystyle G}$	${\displaystyle {\tfrac {E}{2(1+\nu )}$	${\displaystyle G}$	${\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }$	${\displaystyle G}$	${\displaystyle G}$
${\displaystyle \nu =\,}$	${\displaystyle {\tfrac {3K-E}{6K}$	${\displaystyle {\tfrac {\lambda }{3K-\lambda }$	${\displaystyle {\tfrac {3K-2G}{2(3K+G)}$	${\displaystyle \nu }$	${\displaystyle {\tfrac {E}{2G}-1}$	${\displaystyle \nu }$	${\displaystyle {\tfrac {\lambda }{2(\lambda +G)}$	${\displaystyle \nu }$	${\displaystyle \nu }$	${\displaystyle {\tfrac {M-2G}{2M-2G}$
${\displaystyle M=\,}$	${\displaystyle {\tfrac {3K(3K+E)}{9K-E}$	${\displaystyle 3K-2\lambda \,}$	${\displaystyle K+{\tfrac {4G}{3}$	${\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }$	${\displaystyle {\tfrac {G(4G-E)}{3G-E}$	${\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}$	${\displaystyle \lambda +2G\,}$	${\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }$	${\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }$	${\displaystyle M}$