# Предлошка:Модули на растегливост

Формули за претворање
Својствата на еднородните изотропни растегливи материјали се определуваат со два од овие модули; оттука можат да се пресметаат и останатите согласно дадените формули.
${\displaystyle K=\,}$ ${\displaystyle E=\,}$ ${\displaystyle \lambda =\,}$ ${\displaystyle G=\,}$ ${\displaystyle \nu =\,}$ ${\displaystyle M=\,}$ Напомени
${\displaystyle (K,\,E)}$ ${\displaystyle K}$ ${\displaystyle E}$ ${\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}$ ${\displaystyle {\tfrac {3KE}{9K-E}}}$ ${\displaystyle {\tfrac {3K-E}{6K}}}$ ${\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}$
${\displaystyle (K,\,\lambda )}$ ${\displaystyle K}$ ${\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}$ ${\displaystyle \lambda }$ ${\displaystyle {\tfrac {3(K-\lambda )}{2}}}$ ${\displaystyle {\tfrac {\lambda }{3K-\lambda }}}$ ${\displaystyle 3K-2\lambda \,}$
${\displaystyle (K,\,G)}$ ${\displaystyle K}$ ${\displaystyle {\tfrac {9KG}{3K+G}}}$ ${\displaystyle K-{\tfrac {2G}{3}}}$ ${\displaystyle G}$ ${\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}$ ${\displaystyle K+{\tfrac {4G}{3}}}$
${\displaystyle (K,\,\nu )}$ ${\displaystyle K}$ ${\displaystyle 3K(1-2\nu )\,}$ ${\displaystyle {\tfrac {3K\nu }{1+\nu }}}$ ${\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}$ ${\displaystyle \nu }$ ${\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}$
${\displaystyle (K,\,M)}$ ${\displaystyle K}$ ${\displaystyle {\tfrac {9K(M-K)}{3K+M}}}$ ${\displaystyle {\tfrac {3K-M}{2}}}$ ${\displaystyle {\tfrac {3(M-K)}{4}}}$ ${\displaystyle {\tfrac {3K-M}{3K+M}}}$ ${\displaystyle M}$
${\displaystyle (E,\,\lambda )}$ ${\displaystyle {\tfrac {E+3\lambda +R}{6}}}$ ${\displaystyle E}$ ${\displaystyle \lambda }$ ${\displaystyle {\tfrac {E-3\lambda +R}{4}}}$ ${\displaystyle {\tfrac {2\lambda }{E+\lambda +R}}}$ ${\displaystyle {\tfrac {E-\lambda +R}{2}}}$ ${\displaystyle R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}$
${\displaystyle (E,\,G)}$ ${\displaystyle {\tfrac {EG}{3(3G-E)}}}$ ${\displaystyle E}$ ${\displaystyle {\tfrac {G(E-2G)}{3G-E}}}$ ${\displaystyle G}$ ${\displaystyle {\tfrac {E}{2G}}-1}$ ${\displaystyle {\tfrac {G(4G-E)}{3G-E}}}$
${\displaystyle (E,\,\nu )}$ ${\displaystyle {\tfrac {E}{3(1-2\nu )}}}$ ${\displaystyle E}$ ${\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}$ ${\displaystyle {\tfrac {E}{2(1+\nu )}}}$ ${\displaystyle \nu }$ ${\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}$
${\displaystyle (E,\,M)}$ ${\displaystyle {\tfrac {3M-E+S}{6}}}$ ${\displaystyle E}$ ${\displaystyle {\tfrac {M-E+S}{4}}}$ ${\displaystyle {\tfrac {3M+E-S}{8}}}$ ${\displaystyle {\tfrac {E-M+S}{4M}}}$ ${\displaystyle M}$

${\displaystyle S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}}$

Има две исправни решенија.
Знакот плус дава ${\displaystyle \nu \geq 0}$.

Знакот минус дава ${\displaystyle \nu \leq 0}$.

${\displaystyle (\lambda ,\,G)}$ ${\displaystyle \lambda +{\tfrac {2G}{3}}}$ ${\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}$ ${\displaystyle \lambda }$ ${\displaystyle G}$ ${\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}$ ${\displaystyle \lambda +2G\,}$
${\displaystyle (\lambda ,\,\nu )}$ ${\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}$ ${\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}$ ${\displaystyle \lambda }$ ${\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}$ ${\displaystyle \nu }$ ${\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}$ Не може да се користи кога ${\displaystyle \nu =0\Leftrightarrow \lambda =0}$
${\displaystyle (\lambda ,\,M)}$ ${\displaystyle {\tfrac {M+2\lambda }{3}}}$ ${\displaystyle {\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}}$ ${\displaystyle \lambda }$ ${\displaystyle {\tfrac {M-\lambda }{2}}}$ ${\displaystyle {\tfrac {\lambda }{M+\lambda }}}$ ${\displaystyle M}$
${\displaystyle (G,\,\nu )}$ ${\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}$ ${\displaystyle 2G(1+\nu )\,}$ ${\displaystyle {\tfrac {2G\nu }{1-2\nu }}}$ ${\displaystyle G}$ ${\displaystyle \nu }$ ${\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}$
${\displaystyle (G,\,M)}$ ${\displaystyle M-{\tfrac {4G}{3}}}$ ${\displaystyle {\tfrac {G(3M-4G)}{M-G}}}$ ${\displaystyle M-2G\,}$ ${\displaystyle G}$ ${\displaystyle {\tfrac {M-2G}{2M-2G}}}$ ${\displaystyle M}$
${\displaystyle (\nu ,\,M)}$ ${\displaystyle {\tfrac {M(1+\nu )}{3(1-\nu )}}}$ ${\displaystyle {\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}}$ ${\displaystyle {\tfrac {M\nu }{1-\nu }}}$ ${\displaystyle {\tfrac {M(1-2\nu )}{2(1-\nu )}}}$ ${\displaystyle \nu }$ ${\displaystyle M}$

Матрицата на крутост (9 на 9 или 6 на 6 по Фојгтовиот запис) во Хуковиот закон (во 3Д) може да се параметризираљ според само две составници за еднородни изотропни материјали. Можеме да избереме било кој пар модули. Некои можни претворања се наведени во табелата.

## Наводи

• G. Mavko, T. Mukerji, J. Dvorkin. The Rock Physics Handbook. Cambridge University Press 2003 (paperback). ISBN 0-521-54344-4