v t e Elastic moduli for homogeneous isotropic materials
Bulk modulus ( $K$ ) Young's modulus ( $E$ ) Lamé's first parameter ( $\lambda$ ) Shear modulus ( $G,\mu$ ) Poisson's ratio ( $\nu$ ) P-wave modulus ( $M$ )

Conversion formulae
Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these; thus, given any two, any other of the elastic moduli can be calculated according to these formulas, provided both for 3D materials (first part of the table) and for 2D materials (second part).
3D formulae	$K=\,$	$E=\,$	$\lambda =\,$	$G=\,$	$\nu =\,$	$M=\,$	Notes
$(K,\,E)$			${\tfrac {3K(3K-E)}{9K-E))$	${\tfrac {3KE}{9K-E))$	${\tfrac {3K-E}{6K))$	${\tfrac {3K(3K+E)}{9K-E))$
$(K,\,\lambda )$		${\tfrac {9K(K-\lambda )}{3K-\lambda ))$		${\tfrac {3(K-\lambda )}{2))$	${\tfrac {\lambda }{3K-\lambda ))$	$3K-2\lambda \,$
$(K,\,G)$		${\tfrac {9KG}{3K+G))$	$K-{\tfrac {2G}{3))$		${\tfrac {3K-2G}{2(3K+G)))$	$K+{\tfrac {4G}{3))$
$(K,\,\nu )$		$3K(1-2\nu )\,$	${\tfrac {3K\nu }{1+\nu ))$	${\tfrac {3K(1-2\nu )}{2(1+\nu )))$		${\tfrac {3K(1-\nu )}{1+\nu ))$
$(K,\,M)$		${\tfrac {9K(M-K)}{3K+M))$	${\tfrac {3K-M}{2))$	${\tfrac {3(M-K)}{4))$	${\tfrac {3K-M}{3K+M))$
$(E,\,\lambda )$	${\tfrac {E+3\lambda +R}{6))$			${\tfrac {E-3\lambda +R}{4))$	${\tfrac {2\lambda }{E+\lambda +R))$	${\tfrac {E-\lambda +R}{2))$	$R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda ))$
$(E,\,G)$	${\tfrac {EG}{3(3G-E)))$		${\tfrac {G(E-2G)}{3G-E))$		${\tfrac {E}{2G))-1$	${\tfrac {G(4G-E)}{3G-E))$
$(E,\,\nu )$	${\tfrac {E}{3(1-2\nu )))$		${\tfrac {E\nu }{(1+\nu )(1-2\nu )))$	${\tfrac {E}{2(1+\nu )))$		${\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )))$
$(E,\,M)$	${\tfrac {3M-E+S}{6))$		${\tfrac {M-E+S}{4))$	${\tfrac {3M+E-S}{8))$	${\tfrac {E-M+S}{4M))$		$S=\pm {\sqrt {E^{2}+9M^{2}-10EM))$ There are two valid solutions. The plus sign leads to $\nu \geq 0$ . The minus sign leads to $\nu \leq 0$ .
$(\lambda ,\,G)$	$\lambda +{\tfrac {2G}{3))$	${\tfrac {G(3\lambda +2G)}{\lambda +G))$			${\tfrac {\lambda }{2(\lambda +G)))$	$\lambda +2G\,$
$(\lambda ,\,\nu )$	${\tfrac {\lambda (1+\nu )}{3\nu ))$	${\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu ))$		${\tfrac {\lambda (1-2\nu )}{2\nu ))$		${\tfrac {\lambda (1-\nu )}{\nu ))$	Cannot be used when $\nu =0\Leftrightarrow \lambda =0$
$(\lambda ,\,M)$	${\tfrac {M+2\lambda }{3))$	${\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda ))$		${\tfrac {M-\lambda }{2))$	${\tfrac {\lambda }{M+\lambda ))$
$(G,\,\nu )$	${\tfrac {2G(1+\nu )}{3(1-2\nu )))$	$2G(1+\nu )\,$	${\tfrac {2G\nu }{1-2\nu ))$			${\tfrac {2G(1-\nu )}{1-2\nu ))$
$(G,\,M)$	$M-{\tfrac {4G}{3))$	${\tfrac {G(3M-4G)}{M-G))$	$M-2G\,$		${\tfrac {M-2G}{2M-2G))$
$(\nu ,\,M)$	${\tfrac {M(1+\nu )}{3(1-\nu )))$	${\tfrac {M(1+\nu )(1-2\nu )}{1-\nu ))$	${\tfrac {M\nu }{1-\nu ))$	${\tfrac {M(1-2\nu )}{2(1-\nu )))$
2D formulae	$K_{\mathrm {2D} }=\,$	$E_{\mathrm {2D} }=\,$	$\lambda _{\mathrm {2D} }=\,$	$G_{\mathrm {2D} }=\,$	$\nu _{\mathrm {2D} }=\,$	$M_{\mathrm {2D} }=\,$	Notes
$(K_{\mathrm {2D} },\,E_{\mathrm {2D} })$			${\displaystyle {\tfrac {2K_{\mathrm {2D} }(2K_{\mathrm {2D} }-E_{\mathrm {2D} })}{4K_{\mathrm {2D} }-E_{\mathrm {2D} ))))$	${\displaystyle {\tfrac {K_{\mathrm {2D} }E_{\mathrm {2D} )){4K_{\mathrm {2D} }-E_{\mathrm {2D} ))))$	${\displaystyle {\tfrac {2K_{\mathrm {2D} }-E_{\mathrm {2D} )){2K_{\mathrm {2D} ))))$	${\displaystyle {\tfrac {4K_{\mathrm {2D} }^{2)){4K_{\mathrm {2D} }-E_{\mathrm {2D} ))))$
$(K_{\mathrm {2D} },\,\lambda _{\mathrm {2D} })$		${\displaystyle {\tfrac {4K_{\mathrm {2D} }(K_{\mathrm {2D} }-\lambda _{\mathrm {2D} })}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} ))))$		${\displaystyle K_{\mathrm {2D} }-\lambda _{\mathrm {2D} ))$	${\displaystyle {\tfrac {\lambda _{\mathrm {2D} )){2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} ))))$	${\displaystyle 2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} ))$
$(K_{\mathrm {2D} },\,G_{\mathrm {2D} })$		${\displaystyle {\tfrac {4K_{\mathrm {2D} }G_{\mathrm {2D} )){K_{\mathrm {2D} }+G_{\mathrm {2D} ))))$	${\displaystyle K_{\mathrm {2D} }-G_{\mathrm {2D} ))$		${\displaystyle {\tfrac {K_{\mathrm {2D} }-G_{\mathrm {2D} )){K_{\mathrm {2D} }+G_{\mathrm {2D} ))))$	${\displaystyle K_{\mathrm {2D} }+G_{\mathrm {2D} ))$
$(K_{\mathrm {2D} },\,\nu _{\mathrm {2D} })$		$2K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })\,$	${\displaystyle {\tfrac {2K_{\mathrm {2D} }\nu _{\mathrm {2D} )){1+\nu _{\mathrm {2D} ))))$	${\displaystyle {\tfrac {K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{1+\nu _{\mathrm {2D} ))))$		${\displaystyle {\tfrac {2K_{\mathrm {2D} )){1+\nu _{\mathrm {2D} ))))$
$(E_{\mathrm {2D} },\,G_{\mathrm {2D} })$	${\displaystyle {\tfrac {E_{\mathrm {2D} }G_{\mathrm {2D} )){4G_{\mathrm {2D} }-E_{\mathrm {2D} ))))$		${\displaystyle {\tfrac {2G_{\mathrm {2D} }(E_{\mathrm {2D} }-2G_{\mathrm {2D} })}{4G_{\mathrm {2D} }-E_{\mathrm {2D} ))))$		${\tfrac {E_{\mathrm {2D} )){2G_{\mathrm {2D} ))}-1$	${\displaystyle {\tfrac {4G_{\mathrm {2D} }^{2)){4G_{\mathrm {2D} }-E_{\mathrm {2D} ))))$
$(E_{\mathrm {2D} },\,\nu _{\mathrm {2D} })$	${\tfrac {E_{\mathrm {2D} )){2(1-\nu _{\mathrm {2D} })))$		${\tfrac {E_{\mathrm {2D} }\nu _{\mathrm {2D} )){(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })))$	${\tfrac {E_{\mathrm {2D} )){2(1+\nu _{\mathrm {2D} })))$		${\tfrac {E_{\mathrm {2D} )){(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })))$
$(\lambda _{\mathrm {2D} },\,G_{\mathrm {2D} })$	${\displaystyle \lambda _{\mathrm {2D} }+G_{\mathrm {2D} ))$	${\displaystyle {\tfrac {4G_{\mathrm {2D} }(\lambda _{\mathrm {2D} }+G_{\mathrm {2D} })}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} ))))$			${\displaystyle {\tfrac {\lambda _{\mathrm {2D} )){\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} ))))$	$\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }\,$
$(\lambda _{\mathrm {2D} },\,\nu _{\mathrm {2D} })$	${\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} ))))$	${\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}{\nu _{\mathrm {2D} ))))$		${\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} ))))$		${\displaystyle {\tfrac {\lambda _{\mathrm {2D} )){\nu _{\mathrm {2D} ))))$	Cannot be used when $\nu _{\mathrm {2D} }=0\Leftrightarrow \lambda _{\mathrm {2D} }=0$
$(G_{\mathrm {2D} },\,\nu _{\mathrm {2D} })$	${\displaystyle {\tfrac {G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{1-\nu _{\mathrm {2D} ))))$	$2G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })\,$	${\displaystyle {\tfrac {2G_{\mathrm {2D} }\nu _{\mathrm {2D} )){1-\nu _{\mathrm {2D} ))))$			${\displaystyle {\tfrac {2G_{\mathrm {2D} )){1-\nu _{\mathrm {2D} ))))$
$(G_{\mathrm {2D} },\,M_{\mathrm {2D} })$	${\displaystyle M_{\mathrm {2D} }-G_{\mathrm {2D} ))$	${\displaystyle {\tfrac {4G_{\mathrm {2D} }(M_{\mathrm {2D} }-G_{\mathrm {2D} })}{M_{\mathrm {2D} ))))$	$M_{\mathrm {2D} }-2G_{\mathrm {2D} }\,$		${\displaystyle {\tfrac {M_{\mathrm {2D} }-2G_{\mathrm {2D} )){M_{\mathrm {2D} ))))$

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