# 剪切模量

（重定向自剪力模數

G, S

G = τ / γ

${\displaystyle G={\frac {\tau }{\gamma }}}$

${\displaystyle G={E \over {2(1+\nu )}}}$

## 波

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

G是剪切模量
${\displaystyle \rho }$ 是固体的密度.

## 金属的剪切模量

1. MTS剪切模量模型由机械阈值应力(MTS)塑性流动应力模型开发并与之结合使用。[4][5][6]
2. 由SCGL流动应力模型开发并与之结合使用的SCGL剪切模量模型。[7]
3. 纳达尔和LePoac (NP)剪切模量模型，利用Lindemann理论确定剪切模量对温度的依赖关系，利用SCG模型确定剪切模量对压力的依赖关系。[2]

### MTS剪切模型

MTS剪切模量模型为:

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

## 参见

${\displaystyle (\lambda ,\,G)}$  ${\displaystyle (E,\,G)}$  ${\displaystyle (K,\,\lambda )}$  ${\displaystyle (K,\,G)}$  ${\displaystyle (\lambda ,\,\nu )}$  ${\displaystyle (G,\,\nu )}$  ${\displaystyle (E,\,\nu )}$  ${\displaystyle (K,\,\nu )}$  ${\displaystyle (K,\,E)}$  ${\displaystyle (M,\,G)}$
${\displaystyle K=\,}$  ${\displaystyle \lambda +{\tfrac {2G}{3}}}$  ${\displaystyle {\tfrac {EG}{3(3G-E)}}}$  ${\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}$  ${\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}$  ${\displaystyle {\tfrac {E}{3(1-2\nu )}}}$  ${\displaystyle M-{\tfrac {4G}{3}}}$
${\displaystyle E=\,}$  ${\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}$  ${\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}$  ${\displaystyle {\tfrac {9KG}{3K+G}}}$  ${\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}$  ${\displaystyle 2G(1+\nu )\,}$  ${\displaystyle 3K(1-2\nu )\,}$  ${\displaystyle {\tfrac {G(3M-4G)}{M-G}}}$
${\displaystyle \lambda =\,}$  ${\displaystyle {\tfrac {G(E-2G)}{3G-E}}}$  ${\displaystyle K-{\tfrac {2G}{3}}}$  ${\displaystyle {\tfrac {2G\nu }{1-2\nu }}}$  ${\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}$  ${\displaystyle {\tfrac {3K\nu }{1+\nu }}}$  ${\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}$  ${\displaystyle M-2G\,}$
${\displaystyle G=\,}$  ${\displaystyle {\tfrac {3(K-\lambda )}{2}}}$  ${\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}$  ${\displaystyle {\tfrac {E}{2(1+\nu )}}}$  ${\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}$  ${\displaystyle {\tfrac {3KE}{9K-E}}}$
${\displaystyle \nu =\,}$  ${\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}$  ${\displaystyle {\tfrac {E}{2G}}-1}$  ${\displaystyle {\tfrac {\lambda }{3K-\lambda }}}$  ${\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}$  ${\displaystyle {\tfrac {3K-E}{6K}}}$  ${\displaystyle {\tfrac {M-2G}{2M-2G}}}$
${\displaystyle M=\,}$  ${\displaystyle \lambda +2G\,}$  ${\displaystyle {\tfrac {G(4G-E)}{3G-E}}}$  ${\displaystyle 3K-2\lambda \,}$  ${\displaystyle K+{\tfrac {4G}{3}}}$  ${\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}$  ${\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}$  ${\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}$  ${\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}$  ${\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}$
1. ^ Overton, W.; Gaffney, John. Temperature Variation of the Elastic Constants of Cubic Elements. I. Copper. Physical Review. 1955, 98 (4): 969. Bibcode:1955PhRv...98..969O. doi:10.1103/PhysRev.98.969.
2. Nadal, Marie-Hélène; Le Poac, Philippe. 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. 2003, 93 (5): 2472. Bibcode:2003JAP....93.2472N. doi:10.1063/1.1539913.
3. ^ March, N. H., (1996), Electron Correlation in Molecules and Condensed Phases页面存档备份，存于互联网档案馆）, Springer, ISBN 0-306-44844-0 p. 363
4. ^ Varshni, Y. Temperature Dependence of the Elastic Constants. Physical Review B. 1970, 2 (10): 3952–3958. Bibcode:1970PhRvB...2.3952V. doi:10.1103/PhysRevB.2.3952.
5. ^ Chen, Shuh Rong; Gray, George T. Constitutive behavior of tantalum and tantalum-tungsten alloys (PDF). Metallurgical and Materials Transactions A. 1996, 27 (10): 2994 [2019-11-22]. Bibcode:1996MMTA...27.2994C. doi:10.1007/BF02663849. （原始内容存档 (PDF)于2020-10-01）.
6. ^ Goto, D. M.; Garrett, R. K.; Bingert, J. F.; Chen, S. R.; Gray, G. T. The mechanical threshold stress constitutive-strength model description of HY-100 steel. Metallurgical and Materials Transactions A. 2000, 31 (8): 1985–1996 [2019-11-22]. doi:10.1007/s11661-000-0226-8. （原始内容存档于2017-09-25）.
7. ^ Guinan, M; Steinberg, D. Pressure and temperature derivatives of the isotropic polycrystalline shear modulus for 65 elements. Journal of Physics and Chemistry of Solids. 1974, 35 (11): 1501. Bibcode:1974JPCS...35.1501G. doi:10.1016/S0022-3697(74)80278-7.