Templat:Modulus elastik

l b s Modulus elastik untuk bahan-bahan isotropik homogen
Modulus kompresi ( K {\displaystyle K} $K$ ) Modulus Young ( E {\displaystyle E} $E$ ) Parameter pertama Lamé ( λ {\displaystyle \lambda } $\lambda$ ) Modulus geser ( G {\displaystyle G} $G$ ) Rasio Poisson ( ν {\displaystyle \nu } $\nu$ ) Modulus P-wave ( M {\displaystyle M} $M$ )

Rumus konversi
Bahan-bahan elastik linear isotropik homogen mempunyai sifat-sifat elastik yang secara unik ditentukan oleh dua dari modulus di atas; jadi, dengan mengetahui dua di antaranya, modulus elastik yang lain dapat dihiung menurut rumus-rumus ini.
	K = {\displaystyle K=\,} $K=\,$	E = {\displaystyle E=\,} $E=\,$	λ = {\displaystyle \lambda =\,} $\lambda =\,$	G = {\displaystyle G=\,} $G=\,$	ν = {\displaystyle \nu =\,} $\nu =\,$	M = {\displaystyle M=\,} $M=\,$	Notes
( K , E ) {\displaystyle (K,\,E)} $(K,\,E)$	K {\displaystyle K} $K$	E {\displaystyle E} $E$	3 K ( 3 K − E ) 9 K − E {\displaystyle {\tfrac {3K(3K-E)}{9K-E}}} ${\tfrac {3K(3K-E)}{9K-E}}$	3 K E 9 K − E {\displaystyle {\tfrac {3KE}{9K-E}}} ${\tfrac {3KE}{9K-E}}$	3 K − E 6 K {\displaystyle {\tfrac {3K-E}{6K}}} ${\tfrac {3K-E}{6K}}$	3 K ( 3 K + E ) 9 K − E {\displaystyle {\tfrac {3K(3K+E)}{9K-E}}} ${\tfrac {3K(3K+E)}{9K-E}}$
( K , λ ) {\displaystyle (K,\,\lambda )} $(K,\,\lambda )$	K {\displaystyle K} $K$	9 K ( K − λ ) 3 K − λ {\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}} ${\tfrac {9K(K-\lambda )}{3K-\lambda }}$	λ {\displaystyle \lambda } $\lambda$	3 ( K − λ ) 2 {\displaystyle {\tfrac {3(K-\lambda )}{2}}} ${\tfrac {3(K-\lambda )}{2}}$	λ 3 K − λ {\displaystyle {\tfrac {\lambda }{3K-\lambda }}} ${\tfrac {\lambda }{3K-\lambda }}$	3 K − 2 λ {\displaystyle 3K-2\lambda \,} $3K-2\lambda \,$
( K , G ) {\displaystyle (K,\,G)} $(K,\,G)$	K {\displaystyle K} $K$	9 K G 3 K + G {\displaystyle {\tfrac {9KG}{3K+G}}} ${\tfrac {9KG}{3K+G}}$	K − 2 G 3 {\displaystyle K-{\tfrac {2G}{3}}} $K-{\tfrac {2G}{3}}$	G {\displaystyle G} $G$	3 K − 2 G 2 ( 3 K + G ) {\displaystyle {\tfrac {3K-2G}{2(3K+G)}}} ${\tfrac {3K-2G}{2(3K+G)}}$	K + 4 G 3 {\displaystyle K+{\tfrac {4G}{3}}} $K+{\tfrac {4G}{3}}$
( K , ν ) {\displaystyle (K,\,\nu )} $(K,\,\nu )$	K {\displaystyle K} $K$	3 K ( 1 − 2 ν ) {\displaystyle 3K(1-2\nu )\,} $3K(1-2\nu )\,$	3 K ν 1 + ν {\displaystyle {\tfrac {3K\nu }{1+\nu }}} ${\tfrac {3K\nu }{1+\nu }}$	3 K ( 1 − 2 ν ) 2 ( 1 + ν ) {\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}} ${\tfrac {3K(1-2\nu )}{2(1+\nu )}}$	ν {\displaystyle \nu } $\nu$	3 K ( 1 − ν ) 1 + ν {\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}} ${\tfrac {3K(1-\nu )}{1+\nu }}$
( K , M ) {\displaystyle (K,\,M)} $(K,\,M)$	K {\displaystyle K} $K$	9 K ( M − K ) 3 K + M {\displaystyle {\tfrac {9K(M-K)}{3K+M}}} ${\tfrac {9K(M-K)}{3K+M}}$	3 K − M 2 {\displaystyle {\tfrac {3K-M}{2}}} ${\tfrac {3K-M}{2}}$	3 ( M − K ) 4 {\displaystyle {\tfrac {3(M-K)}{4}}} ${\tfrac {3(M-K)}{4}}$	3 K − M 3 K + M {\displaystyle {\tfrac {3K-M}{3K+M}}} ${\tfrac {3K-M}{3K+M}}$	M {\displaystyle M} $M$
( E , λ ) {\displaystyle (E,\,\lambda )} $(E,\,\lambda )$	E + 3 λ + R 6 {\displaystyle {\tfrac {E+3\lambda +R}{6}}} ${\tfrac {E+3\lambda +R}{6}}$	E {\displaystyle E} $E$	λ {\displaystyle \lambda } $\lambda$	E − 3 λ + R 4 {\displaystyle {\tfrac {E-3\lambda +R}{4}}} ${\tfrac {E-3\lambda +R}{4}}$	2 λ E + λ + R {\displaystyle {\tfrac {2\lambda }{E+\lambda +R}}} ${\tfrac {2\lambda }{E+\lambda +R}}$	E − λ + R 2 {\displaystyle {\tfrac {E-\lambda +R}{2}}} ${\tfrac {E-\lambda +R}{2}}$	R = E 2 + 9 λ 2 + 2 E λ {\displaystyle R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}} $R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}$
( E , G ) {\displaystyle (E,\,G)} $(E,\,G)$	E G 3 ( 3 G − E ) {\displaystyle {\tfrac {EG}{3(3G-E)}}} ${\tfrac {EG}{3(3G-E)}}$	E {\displaystyle E} $E$	G ( E − 2 G ) 3 G − E {\displaystyle {\tfrac {G(E-2G)}{3G-E}}} ${\tfrac {G(E-2G)}{3G-E}}$	G {\displaystyle G} $G$	E 2 G − 1 {\displaystyle {\tfrac {E}{2G}}-1} ${\tfrac {E}{2G}}-1$	G ( 4 G − E ) 3 G − E {\displaystyle {\tfrac {G(4G-E)}{3G-E}}} ${\tfrac {G(4G-E)}{3G-E}}$
( E , ν ) {\displaystyle (E,\,\nu )} $(E,\,\nu )$	E 3 ( 1 − 2 ν ) {\displaystyle {\tfrac {E}{3(1-2\nu )}}} ${\tfrac {E}{3(1-2\nu )}}$	E {\displaystyle E} $E$	E ν ( 1 + ν ) ( 1 − 2 ν ) {\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}} ${\tfrac {E\nu }{(1+\nu )(1-2\nu )}}$	E 2 ( 1 + ν ) {\displaystyle {\tfrac {E}{2(1+\nu )}}} ${\tfrac {E}{2(1+\nu )}}$	ν {\displaystyle \nu } $\nu$	E ( 1 − ν ) ( 1 + ν ) ( 1 − 2 ν ) {\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}} ${\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}$
( E , M ) {\displaystyle (E,\,M)} $(E,\,M)$	3 M − E + S 6 {\displaystyle {\tfrac {3M-E+S}{6}}} ${\tfrac {3M-E+S}{6}}$	E {\displaystyle E} $E$	M − E + S 4 {\displaystyle {\tfrac {M-E+S}{4}}} ${\tfrac {M-E+S}{4}}$	3 M + E − S 8 {\displaystyle {\tfrac {3M+E-S}{8}}} ${\tfrac {3M+E-S}{8}}$	E − M + S 4 M {\displaystyle {\tfrac {E-M+S}{4M}}} ${\tfrac {E-M+S}{4M}}$	M {\displaystyle M} $M$	S = ± E 2 + 9 M 2 − 10 E M {\displaystyle S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}} $S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}$ Ada dua pemecahan valid. Tanda plus mengarah kepada ν ≥ 0 {\displaystyle \nu \geq 0} $\nu \geq 0$ . Tanda minus mengarah kepada ν ≤ 0 {\displaystyle \nu \leq 0} $\nu \leq 0$ .
( λ , G ) {\displaystyle (\lambda ,\,G)} $(\lambda ,\,G)$	λ + 2 G 3 {\displaystyle \lambda +{\tfrac {2G}{3}}} $\lambda +{\tfrac {2G}{3}}$	G ( 3 λ + 2 G ) λ + G {\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}} ${\tfrac {G(3\lambda +2G)}{\lambda +G}}$	λ {\displaystyle \lambda } $\lambda$	G {\displaystyle G} $G$	λ 2 ( λ + G ) {\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}} ${\tfrac {\lambda }{2(\lambda +G)}}$	λ + 2 G {\displaystyle \lambda +2G\,} $\lambda +2G\,$
( λ , ν ) {\displaystyle (\lambda ,\,\nu )} $(\lambda ,\,\nu )$	λ ( 1 + ν ) 3 ν {\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}} ${\tfrac {\lambda (1+\nu )}{3\nu }}$	λ ( 1 + ν ) ( 1 − 2 ν ) ν {\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}} ${\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}$	λ {\displaystyle \lambda } $\lambda$	λ ( 1 − 2 ν ) 2 ν {\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}} ${\tfrac {\lambda (1-2\nu )}{2\nu }}$	ν {\displaystyle \nu } $\nu$	λ ( 1 − ν ) ν {\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}} ${\tfrac {\lambda (1-\nu )}{\nu }}$	Tidak dapat digunakan bilamana ν = 0 ⇔ λ = 0 {\displaystyle \nu =0\Leftrightarrow \lambda =0} $\nu =0\Leftrightarrow \lambda =0$
( λ , M ) {\displaystyle (\lambda ,\,M)} $(\lambda ,\,M)$	M + 2 λ 3 {\displaystyle {\tfrac {M+2\lambda }{3}}} ${\tfrac {M+2\lambda }{3}}$	( M − λ ) ( M + 2 λ ) M + λ {\displaystyle {\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}} ${\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}$	λ {\displaystyle \lambda } $\lambda$	M − λ 2 {\displaystyle {\tfrac {M-\lambda }{2}}} ${\tfrac {M-\lambda }{2}}$	λ M + λ {\displaystyle {\tfrac {\lambda }{M+\lambda }}} ${\tfrac {\lambda }{M+\lambda }}$	M {\displaystyle M} $M$
( G , ν ) {\displaystyle (G,\,\nu )} $(G,\,\nu )$	2 G ( 1 + ν ) 3 ( 1 − 2 ν ) {\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}} ${\tfrac {2G(1+\nu )}{3(1-2\nu )}}$	2 G ( 1 + ν ) {\displaystyle 2G(1+\nu )\,} $2G(1+\nu )\,$	2 G ν 1 − 2 ν {\displaystyle {\tfrac {2G\nu }{1-2\nu }}} ${\tfrac {2G\nu }{1-2\nu }}$	G {\displaystyle G} $G$	ν {\displaystyle \nu } $\nu$	2 G ( 1 − ν ) 1 − 2 ν {\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}} ${\tfrac {2G(1-\nu )}{1-2\nu }}$
( G , M ) {\displaystyle (G,\,M)} $(G,\,M)$	M − 4 G 3 {\displaystyle M-{\tfrac {4G}{3}}} $M-{\tfrac {4G}{3}}$	G ( 3 M − 4 G ) M − G {\displaystyle {\tfrac {G(3M-4G)}{M-G}}} ${\tfrac {G(3M-4G)}{M-G}}$	M − 2 G {\displaystyle M-2G\,} $M-2G\,$	G {\displaystyle G} $G$	M − 2 G 2 M − 2 G {\displaystyle {\tfrac {M-2G}{2M-2G}}} ${\tfrac {M-2G}{2M-2G}}$	M {\displaystyle M} $M$
( ν , M ) {\displaystyle (\nu ,\,M)} $(\nu ,\,M)$	M ( 1 + ν ) 3 ( 1 − ν ) {\displaystyle {\tfrac {M(1+\nu )}{3(1-\nu )}}} ${\tfrac {M(1+\nu )}{3(1-\nu )}}$	M ( 1 + ν ) ( 1 − 2 ν ) 1 − ν {\displaystyle {\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}} ${\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}$	M ν 1 − ν {\displaystyle {\tfrac {M\nu }{1-\nu }}} ${\tfrac {M\nu }{1-\nu }}$	M ( 1 − 2 ν ) 2 ( 1 − ν ) {\displaystyle {\tfrac {M(1-2\nu )}{2(1-\nu )}}} ${\tfrac {M(1-2\nu )}{2(1-\nu )}}$	ν {\displaystyle \nu } $\nu$	M {\displaystyle M} $M$

Matriks kekakuan (9 x 9, atau 6 x 6 dalam notasi Voigt) pada Hukum Hooke (dalam 3D) dapat dijadikan parameter hanya dengan dua komponen untuk bahan homogen dan isotropik. Dapat dipilih pasangan mana yang lebih disukai di antara modulus elastik tersebut. Sejumlah konversi yang mungkin dilakukan telah dimuat dalam tabel.