참고 공식
균질한 등방성 선형 탄성 재료는 상술한 탄성 계수들 중 두 개로 고유하게 결정되는 탄성 특성을 갖는다. 따라서, 두 개의 탄성 계수만 알고 있으면 나머지는 후술할 공식들로 계산할 수 있다.
K = {\displaystyle K=\,}
E = {\displaystyle E=\,}
λ = {\displaystyle \lambda =\,}
G = {\displaystyle G=\,}
ν = {\displaystyle \nu =\,}
M = {\displaystyle M=\,}
비고
( K , E ) {\displaystyle (K,\,E)}
3 K ( 3 K − E ) 9 K − E {\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}
3 K E 9 K − E {\displaystyle {\tfrac {3KE}{9K-E}}}
3 K − E 6 K {\displaystyle {\tfrac {3K-E}{6K}}}
3 K ( 3 K + E ) 9 K − E {\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}
( K , λ ) {\displaystyle (K,\,\lambda )}
9 K ( K − λ ) 3 K − λ {\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}
3 ( K − λ ) 2 {\displaystyle {\tfrac {3(K-\lambda )}{2}}}
λ 3 K − λ {\displaystyle {\tfrac {\lambda }{3K-\lambda }}}
3 K − 2 λ {\displaystyle 3K-2\lambda \,}
( K , G ) {\displaystyle (K,\,G)}
9 K G 3 K + G {\displaystyle {\tfrac {9KG}{3K+G}}}
K − 2 G 3 {\displaystyle K-{\tfrac {2G}{3}}}
3 K − 2 G 2 ( 3 K + G ) {\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}
K + 4 G 3 {\displaystyle K+{\tfrac {4G}{3}}}
( K , ν ) {\displaystyle (K,\,\nu )}
3 K ( 1 − 2 ν ) {\displaystyle 3K(1-2\nu )\,}
3 K ν 1 + ν {\displaystyle {\tfrac {3K\nu }{1+\nu }}}
3 K ( 1 − 2 ν ) 2 ( 1 + ν ) {\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}
3 K ( 1 − ν ) 1 + ν {\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}
( K , M ) {\displaystyle (K,\,M)}
9 K ( M − K ) 3 K + M {\displaystyle {\tfrac {9K(M-K)}{3K+M}}}
3 K − M 2 {\displaystyle {\tfrac {3K-M}{2}}}
3 ( M − K ) 4 {\displaystyle {\tfrac {3(M-K)}{4}}}
3 K − M 3 K + M {\displaystyle {\tfrac {3K-M}{3K+M}}}
( E , λ ) {\displaystyle (E,\,\lambda )}
E + 3 λ + R 6 {\displaystyle {\tfrac {E+3\lambda +R}{6}}}
E − 3 λ + R 4 {\displaystyle {\tfrac {E-3\lambda +R}{4}}}
2 λ E + λ + R {\displaystyle {\tfrac {2\lambda }{E+\lambda +R}}}
E − λ + R 2 {\displaystyle {\tfrac {E-\lambda +R}{2}}}
R = E 2 + 9 λ 2 + 2 E λ {\displaystyle R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}
( E , G ) {\displaystyle (E,\,G)}
E G 3 ( 3 G − E ) {\displaystyle {\tfrac {EG}{3(3G-E)}}}
G ( E − 2 G ) 3 G − E {\displaystyle {\tfrac {G(E-2G)}{3G-E}}}
E 2 G − 1 {\displaystyle {\tfrac {E}{2G}}-1}
G ( 4 G − E ) 3 G − E {\displaystyle {\tfrac {G(4G-E)}{3G-E}}}
( E , ν ) {\displaystyle (E,\,\nu )}
E 3 ( 1 − 2 ν ) {\displaystyle {\tfrac {E}{3(1-2\nu )}}}
E ν ( 1 + ν ) ( 1 − 2 ν ) {\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}
E 2 ( 1 + ν ) {\displaystyle {\tfrac {E}{2(1+\nu )}}}
E ( 1 − ν ) ( 1 + ν ) ( 1 − 2 ν ) {\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}
( E , M ) {\displaystyle (E,\,M)}
3 M − E + S 6 {\displaystyle {\tfrac {3M-E+S}{6}}}
M − E + S 4 {\displaystyle {\tfrac {M-E+S}{4}}}
3 M + E − S 8 {\displaystyle {\tfrac {3M+E-S}{8}}}
E − M + S 4 M {\displaystyle {\tfrac {E-M+S}{4M}}}
S = ± E 2 + 9 M 2 − 10 E M {\displaystyle S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}} 유효한 해는 두 개다. +은 ν ≥ 0 {\displaystyle \nu \geq 0} 을 유도한다.
−은 ν ≤ 0 {\displaystyle \nu \leq 0} 을 유도한다.
( λ , G ) {\displaystyle (\lambda ,\,G)}
λ + 2 G 3 {\displaystyle \lambda +{\tfrac {2G}{3}}}
G ( 3 λ + 2 G ) λ + G {\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}
λ 2 ( λ + G ) {\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}
λ + 2 G {\displaystyle \lambda +2G\,}
( λ , ν ) {\displaystyle (\lambda ,\,\nu )}
λ ( 1 + ν ) 3 ν {\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}
λ ( 1 + ν ) ( 1 − 2 ν ) ν {\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}
λ ( 1 − 2 ν ) 2 ν {\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}
λ ( 1 − ν ) ν {\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}
ν = 0 ⇔ λ = 0 {\displaystyle \nu =0\Leftrightarrow \lambda =0} 일 때는 사용할 수 없다.
( λ , M ) {\displaystyle (\lambda ,\,M)}
M + 2 λ 3 {\displaystyle {\tfrac {M+2\lambda }{3}}}
( M − λ ) ( M + 2 λ ) M + λ {\displaystyle {\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}}
M − λ 2 {\displaystyle {\tfrac {M-\lambda }{2}}}
λ M + λ {\displaystyle {\tfrac {\lambda }{M+\lambda }}}
( G , ν ) {\displaystyle (G,\,\nu )}
2 G ( 1 + ν ) 3 ( 1 − 2 ν ) {\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}
2 G ( 1 + ν ) {\displaystyle 2G(1+\nu )\,}
2 G ν 1 − 2 ν {\displaystyle {\tfrac {2G\nu }{1-2\nu }}}
2 G ( 1 − ν ) 1 − 2 ν {\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}
( G , M ) {\displaystyle (G,\,M)}
M − 4 G 3 {\displaystyle M-{\tfrac {4G}{3}}}
G ( 3 M − 4 G ) M − G {\displaystyle {\tfrac {G(3M-4G)}{M-G}}}
M − 2 G {\displaystyle M-2G\,}
M − 2 G 2 M − 2 G {\displaystyle {\tfrac {M-2G}{2M-2G}}}
( ν , M ) {\displaystyle (\nu ,\,M)}
M ( 1 + ν ) 3 ( 1 − ν ) {\displaystyle {\tfrac {M(1+\nu )}{3(1-\nu )}}}
M ( 1 + ν ) ( 1 − 2 ν ) 1 − ν {\displaystyle {\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}}
M ν 1 − ν {\displaystyle {\tfrac {M\nu }{1-\nu }}}
M ( 1 − 2 ν ) 2 ( 1 − ν ) {\displaystyle {\tfrac {M(1-2\nu )}{2(1-\nu )}}}