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维基百科自由的百科全书
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剪切模量

维基百科，自由的百科全书
剪力模數
常見符號
G,S
国际单位帕斯卡
從其他物理量的推衍
G =τ /γ
因次L−1MT−2{\displaystyle {\mathsf {L}}^{-1}{\mathsf {M}}{\mathsf {T}}^{-2}}
剪應變

剪力模數（shear modulus）是材料力學中的名詞，彈性材料承受剪應力時會產生剪應變，定義為剪應力與剪應變的比值。公式記為

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

其中，G{\displaystyle G\,} 表示剪力模數，τ{\displaystyle \tau \,} 表示剪應力，γ{\displaystyle \gamma \,} 表示剪應變。在均質且等向性的材料中：

G=E2(1+ν){\displaystyle G={E \over {2(1+\nu )}}}

其中，E{\displaystyle E\,} 是楊氏模數(Young's modulus )，ν{\displaystyle \nu \,} 是泊松比(Poisson's ratio)。

波

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在均匀各向同性固体中，有两种波:P波和S波。剪切波的速度，(vs){\displaystyle (v_{s})}由剪切模量控制，

vs=Gρ{\displaystyle v_{s}={\sqrt {\frac {G}{\rho }}}}

其中

G是剪切模量
ρ{\displaystyle \rho }是固体的密度.

金属的剪切模量

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Shear modulus of copper as a function of temperature. The experimental data[1][2] are shown with colored symbols.

金属的剪切模量通常随温度的升高而降低。在高压下，剪切模量也随外加压力的增大而增大。在许多金属中，熔点温度、空位形成能和剪切模量之间的关系已经被观察到。[3]

有几种模型试图预测金属的剪切模量(可能还有合金的剪切模量)。在塑性流动计算中使用的剪切模量模型包括:

  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

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MTS剪切模量模型为:

μ(T)=μ0−Dexp⁡(T0/T)−1{\displaystyle \mu (T)=\mu _{0}-{\frac {D}{\exp(T_{0}/T)-1}}}

其中μ0{\displaystyle \mu _{0}}为T=0K{\displaystyle T=0K}处的剪切模量，D{\displaystyle D}和T0{\displaystyle T_{0}}为材料常数。

SCG剪切模型

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NP剪切模型

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剪切松弛模量

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参见

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基本定律
固体力学
流体力学
流变学
科學家
均质各向同性材料的彈性模數
换算公式
均质各向同性线弹性材料具有独特的弹性性质，因此知道弹性模量中的任意两种，就可由下列换算公式求出其他所有的弹性模量。
(λ,G){\displaystyle (\lambda ,\,G)}(E,G){\displaystyle (E,\,G)}(K,λ){\displaystyle (K,\,\lambda )}(K,G){\displaystyle (K,\,G)}(λ,ν){\displaystyle (\lambda ,\,\nu )}(G,ν){\displaystyle (G,\,\nu )}(E,ν){\displaystyle (E,\,\nu )}(K,ν){\displaystyle (K,\,\nu )}(K,E){\displaystyle (K,\,E)}(M,G){\displaystyle (M,\,G)}
K={\displaystyle K=\,}λ+2G3{\displaystyle \lambda +{\tfrac {2G}{3}}}EG3(3G−E){\displaystyle {\tfrac {EG}{3(3G-E)}}}λ(1+ν)3ν{\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}2G(1+ν)3(1−2ν){\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}E3(1−2ν){\displaystyle {\tfrac {E}{3(1-2\nu )}}}M−4G3{\displaystyle M-{\tfrac {4G}{3}}}
E={\displaystyle E=\,}G(3λ+2G)λ+G{\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}9K(K−λ)3K−λ{\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}9KG3K+G{\displaystyle {\tfrac {9KG}{3K+G}}}λ(1+ν)(1−2ν)ν{\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}2G(1+ν){\displaystyle 2G(1+\nu )\,}3K(1−2ν){\displaystyle 3K(1-2\nu )\,}G(3M−4G)M−G{\displaystyle {\tfrac {G(3M-4G)}{M-G}}}
λ={\displaystyle \lambda =\,}G(E−2G)3G−E{\displaystyle {\tfrac {G(E-2G)}{3G-E}}}K−2G3{\displaystyle K-{\tfrac {2G}{3}}}2Gν1−2ν{\displaystyle {\tfrac {2G\nu }{1-2\nu }}}Eν(1+ν)(1−2ν){\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}3Kν1+ν{\displaystyle {\tfrac {3K\nu }{1+\nu }}}3K(3K−E)9K−E{\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}M−2G{\displaystyle M-2G\,}
G={\displaystyle G=\,}3(K−λ)2{\displaystyle {\tfrac {3(K-\lambda )}{2}}}λ(1−2ν)2ν{\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}E2(1+ν){\displaystyle {\tfrac {E}{2(1+\nu )}}}3K(1−2ν)2(1+ν){\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}3KE9K−E{\displaystyle {\tfrac {3KE}{9K-E}}}
ν={\displaystyle \nu =\,}λ2(λ+G){\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}E2G−1{\displaystyle {\tfrac {E}{2G}}-1}λ3K−λ{\displaystyle {\tfrac {\lambda }{3K-\lambda }}}3K−2G2(3K+G){\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}3K−E6K{\displaystyle {\tfrac {3K-E}{6K}}}M−2G2M−2G{\displaystyle {\tfrac {M-2G}{2M-2G}}}
M={\displaystyle M=\,}λ+2G{\displaystyle \lambda +2G\,}G(4G−E)3G−E{\displaystyle {\tfrac {G(4G-E)}{3G-E}}}3K−2λ{\displaystyle 3K-2\lambda \,}K+4G3{\displaystyle K+{\tfrac {4G}{3}}}λ(1−ν)ν{\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}2G(1−ν)1−2ν{\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}E(1−ν)(1+ν)(1−2ν){\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}3K(1−ν)1+ν{\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}3K(3K+E)9K−E{\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. ^2.02.1Nadal, 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. 
检索自“https://zh.wikipedia.org/w/index.php?title=剪切模量&oldid=85732488”
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