Modulus geser

Modulus geser Shear modulus
Simbol umum	G
Satuan SI	pascal
Turunan dari besaran lainnya	G = τ / γ

Modulus geser (bahasa Inggris:shear modulus atau modulus of rigidity) dalam sains bahan, dilambangkan dengan G, atau kadang kala S atau μ, didefinisikan sebagai rasio tegangan geser terhadap regangan geser:^[1]

G = d e f τ x y γ x y = F / A Δ x / l = F l A Δ x {\displaystyle G\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\tau _{xy}}{\gamma _{xy}}}={\frac {F/A}{\Delta x/l}}={\frac {Fl}{A\Delta x}}} $G\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\tau _{xy}}{\gamma _{xy}}}={\frac {F/A}{\Delta x/l}}={\frac {Fl}{A\Delta x}}$

di mana

τ x y = F / A {\displaystyle \tau _{xy}=F/A\,} $\tau _{xy}=F/A\,$ = tegangan geser;

F {\displaystyle F} $F$ adalah gaya yang bekerja

A {\displaystyle A} $A$ adalah luas di mana gaya itu bekerja

dalam teknik, γ x y = Δ x / l = tan ⁡ θ {\displaystyle \gamma _{xy}=\Delta x/l=\tan \theta \,} $\gamma _{xy}=\Delta x/l=\tan \theta \,$ = regangan geser. Selain dari itu, γ x y = θ {\displaystyle \gamma _{xy}=\theta } $\gamma _{xy}=\theta$

Δ x {\displaystyle \Delta x} $\Delta x$ adalah perpindahan transvers

l {\displaystyle l} $l$ adalah panjang awal

Satuan turunan SI modulus geser adalah pascal (Pa), meskipun biasanya dinyatakan dalam gigapascal (GPa) atau dalam ribuan pounds per square inch (ksi). Bentuk dimensional adalah M¹L⁻¹T⁻².

Modulus geser selalu bernilai positif.

Penjelasan

Bahan	Nilai umum untuk modulus geser (GPa) (pada suhu ruangan)
Berlian^[2]	478,0
Baja^[3]	79,3
Tembaga^[4]	44,7
Titanium^[3]	41,4
Kaca^[3]	26,2
Aluminium^[3]	25,5
Polietilena^[3]	0,117
Karet^[5]	0,0006

Modulus geser adalah satu dari beberapa kuantitas untuk pengukuran kekakuan suatu bahan. Semuanya bermula dari generalisasi Hukum Hooke:

Modulus Young menyatakan respons suatu bahan terhadap tegangan linear (seperti menarik ujung suatu kawat atau meletakkan suatu berat di atas sebuah tiang),
Modulus kompresi menyatakan respons suatu bahan terhadap tekanan uniform (seperti tekanan pada dasar samudra atau kolam renang yang dalam)
Modulus geser menyatakan respons suatu bahan terhadap tegangan geser (seperti memotong sesuatu dengan gunting yang tumpul).

Gelombang

Dalam benda padat homogene dan isotropik, ada dua jenis gelombang, gelombang tekanan dan gelombang geser. Kecepatan suatu gelombang geser, ( v s ) {\displaystyle (v_{s})} $(v_{s})$ , dikontrol modulus geser,

v s = G ρ {\displaystyle v_{s}={\sqrt {\frac {G}{\rho }}}} $v_{s}={\sqrt {\frac {G}{\rho }}}$

di mana

G adalah modulus geser

ρ {\displaystyle \rho } $\rho$ adalah densitas benda padat.

Modulus geser logam

Modulus geser logam biasanya diamati menurun seiring dengan naiknya suhu. Pada tekanan tinggi, modulus geser tampaknya juga meningkat seiring dengan meningkatnya tekanan yang diberikan. Korelasi antara titik leleh, energi pembentukan vakansi, dan modulus geser telah diamati pada banyak logam.^[8]

Ada beberapa model yang mencoba meramalkan modulus geser logam (dan juga alloy). Model-model modulus geser yang sudah digunakan dalam komputasi aliran plastik termasuk:

Model modulus geser MTS yang dikembangkan oleh^[9] dan digunakan dalam hubungan dengan model tegangan aliran plastik "Mechanical Threshold Stress" (MTS).^[10]^[11]
Model modulus geser "Steinberg-Cochran-Guinan" (SCG) yang dikembangkan oleh^[12] dan digunakan dalam hubungan dengan model tegangan aliran "Steinberg-Cochran-Guinan-Lund" (SCGL).
Model modulus geser "Nadal and LePoac" (NP)^[7] yang menggunakan teori Lindemann untuk menentukan ketergantungan akan suhu dan model SCG untuk ketergantungan akan tekanan dari modulus geser.

Model modulus geser MTS

Model modulus geser MTS mempunyai bentuk:

μ ( T ) = μ 0 − D exp ⁡ ( T 0 / T ) − 1 {\displaystyle \mu (T)=\mu _{0}-{\frac {D}{\exp(T_{0}/T)-1}}} $\mu (T)=\mu _{0}-{\frac {D}{\exp(T_{0}/T)-1}}$

di mana µ₀ adalah modulus geser pada suhu 0 K, dan D serta T₀ adalah konstanta-konstanta bahan.

Model modulus geser SCG

Model modulus geser Steinberg-Cochran-Guinan (SCG) tergantung pada tekanan dan mempunyai bentuk

μ ( p , T ) = μ 0 + ∂ μ ∂ p p η 1 / 3 + ∂ μ ∂ T ( T − 300 ) ; η := ρ / ρ 0 {\displaystyle \mu (p,T)=\mu _{0}+{\frac {\partial \mu }{\partial p}}{\frac {p}{\eta ^{1/3}}}+{\frac {\partial \mu }{\partial T}}(T-300);\quad \eta :=\rho /\rho _{0}} $\mu (p,T)=\mu _{0}+{\frac {\partial \mu }{\partial p}}{\frac {p}{\eta ^{1/3}}}+{\frac {\partial \mu }{\partial T}}(T-300);\quad \eta :=\rho /\rho _{0}$

di mana, µ₀ adalah modulus geser pada status referensi (reference state; T = 300 K, p = 0, η = 1), p adalah tekanan, dan T adalah suhu.

Model modulus geser NP

Model modulus geser Nadal-Le Poac (NP) adalah suatu versi modifikasi model SCG. Ketergantungan modulus geser secara empiris terhadap suhu pada SCG model digantikan dengan suatu persamaan yang berdasarkan pada teori peleburan Lindemann. Model modulus geser NP mempunyai bentuk:

μ ( p , T ) = 1 J ( T ^ ) [ ( μ 0 + ∂ μ ∂ p p η 1 / 3 ) ( 1 − T ^ ) + ρ C m k b T ] ; C := ( 6 π 2 ) 2 / 3 3 f 2 {\displaystyle \mu (p,T)={\frac {1}{{\mathcal {J}}({\hat {T}})}}\left[\left(\mu _{0}+{\frac {\partial \mu }{\partial p}}{\cfrac {p}{\eta ^{1/3}}}\right)(1-{\hat {T}})+{\frac {\rho }{Cm}}~k_{b}~T\right];\quad C:={\cfrac {(6\pi ^{2})^{2/3}}{3}}f^{2}} $\mu (p,T)={\frac {1}{{\mathcal {J}}({\hat {T}})}}\left[\left(\mu _{0}+{\frac {\partial \mu }{\partial p}}{\cfrac {p}{\eta ^{1/3}}}\right)(1-{\hat {T}})+{\frac {\rho }{Cm}}~k_{b}~T\right];\quad C:={\cfrac {(6\pi ^{2})^{2/3}}{3}}f^{2}$

di mana

J ( T ^ ) := 1 + exp ⁡ [ − 1 + 1 / ζ 1 + ζ / ( 1 − T ^ ) ] for T ^ := T T m ∈ [ 0 , 1 + ζ ] , {\displaystyle {\mathcal {J}}({\hat {T}}):=1+\exp \left[-{\cfrac {1+1/\zeta }{1+\zeta /(1-{\hat {T}})}}\right]\quad {\text{for}}\quad {\hat {T}}:={\frac {T}{T_{m}}}\in [0,1+\zeta ],} ${\mathcal {J}}({\hat {T}}):=1+\exp \left[-{\cfrac {1+1/\zeta }{1+\zeta /(1-{\hat {T}})}}\right]\quad {\text{for}}\quad {\hat {T}}:={\frac {T}{T_{m}}}\in [0,1+\zeta ],$

dan µ₀ adalah modulus geser pada suhu 0 K dan tekanan lingkungan, ζ adalah paramater bahan, k_b adalah konstanta Boltzmann, m adalah massa atom, dan f adalah konstanta Lindemann.

Lihat pula

Referensi

↑ IUPAC, Compendium of Chemical Terminology, edisi ke-2 ("Buku Emas") (1997). Versi koreksi daring: (2006–) "shear modulus, G".
↑ McSkimin, H.J.; Andreatch, P. (1972). "Elastic Moduli of Diamond as a Function of Pressure and Temperature". J. Appl. Phys. 43 (7): 2944–2948. Bibcode:1972JAP....43.2944M. doi:10.1063/1.1661636.
1 2 3 4 5 Crandall, Dahl, Lardner (1959). An Introduction to the Mechanics of Solids. Boston: McGraw-Hill. ISBN 0-07-013441-3.{{cite book}}: Ketidakcocokan ISBN / tanggalPemeliharaan CS1: Banyak nama: daftar penulis (link)
↑ "Material properties". Diarsipkan dari asli tanggal 2009-09-01. Diakses tanggal 2014-12-20.
↑ Spanos, Pete (2003). "Cure system effect on low temperature dynamic shear modulus of natural rubber". Rubber World.
↑ Overton, W.; Gaffney, John (1955). "Temperature Variation of the Elastic Constants of Cubic Elements. I. Copper". Physical Review. 98 (4): 969. Bibcode:1955PhRv...98..969O. doi:10.1103/PhysRev.98.969.
1 2 Nadal, Marie-Hélène; Le Poac, Philippe (2003). "Continuous model for the shear modulus as a function of pressure and temperature up to the melting point: Analysis and ultrasonic validation". Journal of Applied Physics. 93 (5): 2472. Bibcode:2003JAP....93.2472N. doi:10.1063/1.1539913.
↑ March, N. H., (1996), Electron Correlation in Molecules and Condensed Phases, Springer, ISBN 0-306-44844-0 p. 363
↑ Varshni, Y. (1970). "Temperature Dependence of the Elastic Constants". Physical Review B. 2 (10): 3952. Bibcode:1970PhRvB...2.3952V. doi:10.1103/PhysRevB.2.3952.
↑ Chen, Shuh Rong; Gray, George T. (1996). "Constitutive behavior of tantalum and tantalum-tungsten alloys". Metallurgical and Materials Transactions A. 27 (10): 2994. Bibcode:1996MMTA...27.2994C. doi:10.1007/BF02663849.
↑ Goto, D. M.; Garrett, R. K.; Bingert, J. F.; Chen, S. R.; Gray, G. T. (2000). "The mechanical threshold stress constitutive-strength model description of HY-100 steel". Metallurgical and Materials Transactions A. 31 (8): 1985–1996. doi:10.1007/s11661-000-0226-8.
↑ Guinan, M; Steinberg, D (1974). "Pressure and temperature derivatives of the isotropic polycrystalline shear modulus for 65 elements". Journal of Physics and Chemistry of Solids. 35 (11): 1501. Bibcode:1974JPCS...35.1501G. doi:10.1016/S0022-3697(74)80278-7.

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} $M$