# साँचा:प्रत्यास्थता मापांक

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

The stiffness matrix (9 by 9, or 6 by 6 in Voigt notation) in Hooke's law (in 3D) can be parametrized by only two components for homogeneous and isotropic materials. One may choose whichever pair one prefers among the elastic moduli given below. Some of the possible conversions are listed in the table.

## पठनीय

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