Modulus kompresi

Modulus kompresi atau modulus curah (bahasa Inggris:bulk modulus; dengan lambang K {\displaystyle K} $K$ atau B {\displaystyle B} $B$ ) suatu zat adalah ukuran resistansi zat itu pada kompresi uniform. Didefinisikan sebagai rasio kenaikan tekanan infinitesimal terhadap penurunan relatif volume yang dihasilkan. Satuan SI modulus kompresi adalah pascal, dan bentuk dimensionalnya adalah M¹L⁻¹T⁻².^[1]

Definisi

Modulus kompresi K > 0 {\displaystyle K>0} $K>0$ dapat secara formal didefinisikan dengan persamaan

K = − V d P d V {\displaystyle K=-V{\frac {\mathrm {d} P}{\mathrm {d} V}}} $K=-V{\frac {\mathrm {d} P}{\mathrm {d} V}}$

di mana P {\displaystyle P} $P$ adalah tekanan, V {\displaystyle V} $V$ adalah volume, dan d P / d V {\displaystyle dP/dV} $dP/dV$ melambangkan turunan tekanan terhadap volume. Secara ekuivalen:

K = ρ d P d ρ {\displaystyle K=\rho {\frac {\mathrm {d} P}{\mathrm {d} \rho }}} $K=\rho {\frac {\mathrm {d} P}{\mathrm {d} \rho }}$

di mana ρ adalah densitas dan dP/dρ melambangkan turunan tekanan terhadap densitas. Invers modulus kompresi adalah kompresibilitas zat tersebut.

Relasi termodinamika

Secara sempit, modulus kompresi adalah suatu kuantitas dari termodinamika, dan untuk memberi spesifikasi suatu modulus kompresi perlu diberi spesifikasi bagaimana suhu berubah-ubah ketika mengalami kompresi: suhu konstan (isotermik K T {\displaystyle K_{T}} $K_{T}$ ), entropi konstan (adiabatik K S {\displaystyle K_{S}} $K_{S}$ ), dan variasi-variasi lain mungkin terjadi. Pembedaan demikian khususnya relevan untuk gas.

Bagi suatu gas ideal, modulus kompresi adiabatik K S {\displaystyle K_{S}} $K_{S}$ dihitung dengan

K S = γ P {\displaystyle K_{S}=\gamma \,P} $K_{S}=\gamma \,P$

dan modulus kompresi isotermal K T {\displaystyle K_{T}} $K_{T}$ dihitung dengan

K T = P {\displaystyle K_{T}=P\,} $K_{T}=P\,$

di mana

γ adalah indeks adiabatik, kadangkala disebut κ.

P adalah tekanan.

Bilamana gas itu bukan ideal, persamaan-persamaan ini hanya memberi perkiraan modulus kompresi. Dalam suatu cairan, modulus kompresi K dan densitas ρ ditentukan oleh kecepatan suara c (pressure waves), menurut rumus Newton-Laplace

c = K ρ . {\displaystyle c={\sqrt {\frac {K}{\rho }}}.} $c={\sqrt {\frac {K}{\rho }}}.$

Dalam benda padat, K S {\displaystyle K_{S}} $K_{S}$ dan K T {\displaystyle K_{T}} $K_{T}$ mempunyai nilai yang hampir sama. Benda padat juga dapat menahan gelombang transvers: untuk bahan-bahan semacam ini satu modulus elastik tamabahan, misalnya modulus geser, dibutuhkan untuk menentukan kecepatan gelombang.

Pengukuran

Modulus kompresi dapat diukur menggunakan difraksi bubuk di bawah tekanan. Ini merupakan sifat suatu cairan yang menunjukkan kemampuan untuk mengubah volume di bawah tekanan.

Sejumlah nilai tertentu

Perkiraan modulus kompresi (K) untuk bahan-bahan umum
Bahan	Modulus kompresi dalam GPa	Modulus kompresi dalam psi
Kaca (lihat juga diagram di bawah tabel)	35 to 55	5,8×10⁶
Baja	160	23×10⁶
Berlian (pada 4K) ^[2]	443	64×10⁶

Suatu bahan dengan modulus kompresi 35 GPa kehilangan satu persen volumenya ketika diberi tekanan eksternal sebesar 0,35 GPa (~3500 Bar).

Perkiraan modulus kompresi (K) untuk bahan-bahan lain
Air	2,2×10⁹ Pa (nilai meningkat pada tekanan yang lebih tinggi)
Metanol	8,23×10⁸ Pa (pada 20 °C dan 1 Atm)
Udara	1,42×10⁵ Pa (modulus kompresi adiabatik)
Udara	1,01×10⁵ Pa (modulus kompresi pada suhu konstan)
Helium padat	5×10⁷ Pa (perkiraan)

Referensi

↑ "Bulk Elastic Properties". hyperphysics. Georgia State University.
↑ Page 52 of "Introduction to Solid State Physics, 8th edition" by Charles Kittel, 2005, ISBN 0-471-41526-X
↑ Fluegel, Alexander. "Bulk modulus calculation of glasses". glassproperties.com.

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} $M$