Shear modulus

material	Typical values ​​for the shear modulus in G Pa (at room temperature)
steel	79.3-81
copper	47
titanium	41.4
Glass	26.2
aluminum	25.5
magnesium	17th
Polyethylene	0.117
rubber	0.0003

Shear modulus of a special basic glass: Influences of the addition of selected glass components

The shear modulus (also slip modulus , G-modulus , shear modulus or torsion modulus ) is a material constant that provides information about the linear-elastic deformation of a component as a result of a shear force or shear stress . The SI unit is Newton per square meter (1 N / m² = 1  Pa ), i.e. the unit of mechanical stress . In material databases , the shear modulus is usually given in N / mm² (= MPa) or kN / mm² (= GPa). ${\displaystyle G}$

In the context of the theory of elasticity , the shear modulus corresponds to the second Lamé constant and bears the symbol there . ${\displaystyle \mu }$

definition

The shear modulus describes the relationship between the shear stress and the tangent of the shear angle (slip): ${\displaystyle \tau }$ ${\displaystyle \gamma }$

${\displaystyle \tau =G\cdot \tan \gamma }$

A first approximation can be used for small angles ( small-angle approximation ). ${\displaystyle \gamma }$ ${\displaystyle \tan \gamma \approx \gamma }$

This formula is analogous to Hooke's law for the 1-axis stress state :

${\displaystyle \sigma =E\cdot \varepsilon }$

When torsional stress of a component to its calculated torsional stiffness of the shear modulus and the torsional constant , which is related to the axis about which the body is twisted: ${\displaystyle I_{\mathrm {T} }}$

${\displaystyle \mathrm {Steifigkeit_{T}} =G\cdot I_{\mathrm {T} },}$

analogous to the determination of the tensile stiffness (from the product of the modulus of elasticity and cross-sectional area).

Relationship with other material constants

In an isotropic material the shear modulus is the elastic modulus  E , the Poisson's ratio  ν  (Poisson's ratio) and the bulk modulus  K in the following relationship:

${\displaystyle G={\frac {1}{2(1+\nu )}}\cdot E={\frac {3KE}{9K-E}}={\frac {3(1-2\nu )}{2(1+\nu )}}\cdot K}$

For linear-elastic, non - auxetic material, the Poisson's number is greater than or equal to zero. The energy conservation results in the positive definiteness of the compression module and the modulus of elasticity . It follows that the Poisson's number is below 0.5. This results in the shear modulus of most materials in the linear-elastic range: ${\displaystyle \left(0\leq \nu <0{,}5\right)}$

${\displaystyle {\frac {1}{3}}E<G\leq {\frac {1}{2}}E}$

Auxetic materials are defined to have a negative Poisson's number, which only a few materials do. Since the shear modulus has a positive definite size due to the conservation of energy, the following applies to auxetic materials in the linear-elastic range:

${\displaystyle {\frac {1}{2}}E<G_{\mathrm {aux} }<+\infty }$

Since the modulus of elasticity is also positive definite, the range of validity results for the Poisson's number ${\displaystyle -1<\nu _{\mathrm {aux} }<0.}$

Conversion between the elastic constants

The module ...	... results from:
	${\displaystyle (K,\,E)}$	${\displaystyle (K,\,\lambda )}$	${\displaystyle (K,\,G)}$	${\displaystyle (K,\,\nu )}$	${\displaystyle (E,\,\lambda )}$	${\displaystyle (E,\,G)}$	${\displaystyle (E,\,\nu )}$	${\displaystyle (\lambda ,\,G)}$	${\displaystyle (\lambda ,\,\nu )}$	${\displaystyle (G,\,\nu )}$	${\displaystyle (G,\,M)}$
Compression module ${\displaystyle K\,}$	${\displaystyle K}$	${\displaystyle K}$	${\displaystyle K}$	${\displaystyle K}$	${\displaystyle (E+3\lambda )+}$ ${\displaystyle {\tfrac {\sqrt {(E+3\lambda )^{2}-4\lambda E}}{6}}}$	${\displaystyle {\tfrac {EG}{3(3G-E)}}}$	${\displaystyle {\tfrac {E}{3(1-2\nu )}}}$	${\displaystyle \lambda +}$ ${\displaystyle {\tfrac {2G}{3}}}$	${\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}$	${\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}$	${\displaystyle M-}$ ${\displaystyle {\tfrac {4G}{3}}}$
modulus of elasticity ${\displaystyle E\,}$	${\displaystyle E}$	${\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}$	${\displaystyle {\tfrac {9KG}{3K+G}}}$	${\displaystyle 3K(1-2\nu )\,}$	${\displaystyle E}$	${\displaystyle E}$	${\displaystyle E}$	${\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}$	${\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}$	${\displaystyle 2G(1+\nu )\,}$	${\displaystyle {\tfrac {G(3M-4G)}{M-G}}}$
1. Lamé constant ${\displaystyle \lambda \,}$	${\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}$	${\displaystyle \lambda }$	${\displaystyle K-}$ ${\displaystyle {\tfrac {2G}{3}}}$	${\displaystyle {\tfrac {3K\nu }{1+\nu }}}$	${\displaystyle \lambda }$	${\displaystyle {\tfrac {G(E-2G)}{3G-E}}}$	${\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}$	${\displaystyle \lambda }$	${\displaystyle \lambda }$	${\displaystyle {\tfrac {2G\nu }{1-2\nu }}}$	${\displaystyle M-2G\,}$
Shear modulus or (2nd Lamé constant) ${\displaystyle G}$ ${\displaystyle \mu }$	${\displaystyle {\tfrac {3KE}{9K-E}}}$	${\displaystyle {\tfrac {3(K-\lambda )}{2}}}$	${\displaystyle G}$	${\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}$	${\displaystyle (E-3\lambda )+}$ ${\displaystyle {\tfrac {\sqrt {(E-3\lambda )^{2}+8\lambda E}}{4}}}$	${\displaystyle G}$	${\displaystyle {\tfrac {E}{2(1+\nu )}}}$	${\displaystyle G}$	${\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}$	${\displaystyle G}$	${\displaystyle G}$
Poisson's number ${\displaystyle \nu \,}$	${\displaystyle {\tfrac {3K-E}{6K}}}$	${\displaystyle {\tfrac {\lambda }{3K-\lambda }}}$	${\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}$	${\displaystyle \nu }$	${\displaystyle -(E+\lambda )+}$ ${\displaystyle {\tfrac {\sqrt {(E+\lambda )^{2}+8\lambda ^{2}}}{4\lambda }}}$	${\displaystyle {\tfrac {E}{2G}}}$ ${\displaystyle -1}$	${\displaystyle \nu }$	${\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}$	${\displaystyle \nu }$	${\displaystyle \nu }$	${\displaystyle {\tfrac {M-2G}{2M-2G}}}$
Longitudinal module ${\displaystyle M\,}$	${\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}$	${\displaystyle 3K-2\lambda \,}$	${\displaystyle K+}$ ${\displaystyle {\tfrac {4G}{3}}}$	${\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}$	${\displaystyle {\tfrac {E-\lambda +{\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}{2}}}$	${\displaystyle {\tfrac {G(4G-E)}{3G-E}}}$	${\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}$	${\displaystyle \lambda +2G\,}$	${\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}$	${\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}$	${\displaystyle M}$

See also

• Complex shear modulus
• Shear strength
• Thrust flow
• Continuum mechanics
• Strength theory
• Rheometer
• Moment magnitude scale

Web links

• Physics internship for beginners. (PDF; 146 kB) Uni Kiel.

Individual evidence

1. ↑ Crandall, Dahl, Lardner: An Introduction to the Mechanics of Solids . McGraw-Hill, Boston 1959.
2. ↑ Eurocode 3: Steel construction. Retrieved May 7, 2020 . 
3. ↑ Calculation of the shear modulus of glasses (English).
4. ↑ G. Mavko, T. Mukerji, J. Dvorkin: The Rock Physics Handbook . Cambridge University Press, 2003, ISBN 0-521-54344-4 (paperback).