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 = τ γ {\displaystyle G={\frac {\tau }{\gamma }}} 快速預覽 剪力模數, 常見符號 ...剪力模數常見符號G, S國際單位帕斯卡從其他物理量的推衍G = τ / γ因次 L − 1 M T − 2 {\displaystyle {\mathsf {L}}^{-1}{\mathsf {M}}{\mathsf {T}}^{-2}} 關閉 剪應變 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 = E 2 ( 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 在均勻各向同性固體中，有兩種波:P波和S波。剪切波的速度， ( v s ) {\displaystyle (v_{s})} 由剪切模量控制， v s = G ρ {\displaystyle v_{s}={\sqrt {\frac {G}{\rho }}}} 其中 G是剪切模量 ρ {\displaystyle \rho } 是固體的密度. 金屬的剪切模量 Shear modulus of copper as a function of temperature. The experimental data[1][2] are shown with colored symbols. 金屬的剪切模量通常隨溫度的升高而降低。在高壓下，剪切模量也隨外加壓力的增大而增大。在許多金屬中，熔點溫度、空位形成能和剪切模量之間的關係已經被觀察到。[3] 有幾種模型試圖預測金屬的剪切模量(可能還有合金的剪切模量)。在塑性流動計算中使用的剪切模量模型包括: MTS剪切模量模型由機械閾值應力(MTS)塑性流動應力模型開發並與之結合使用。[4][5][6] 由SCGL流動應力模型開發並與之結合使用的SCGL剪切模量模型。[7] 納達爾和LePoac (NP)剪切模量模型，利用Lindemann理論確定剪切模量對溫度的依賴關係，利用SCG模型確定剪切模量對壓力的依賴關係。[2] MTS剪切模型 MTS剪切模量模型為: μ ( T ) = μ 0 − D exp ( T 0 / T ) − 1 {\displaystyle \mu (T)=\mu _{0}-{\frac {D}{\exp(T_{0}/T)-1}}} 其中 μ 0 {\displaystyle \mu _{0}} 為 T = 0 K {\displaystyle T=0K} 處的剪切模量， D {\displaystyle D} 和 T 0 {\displaystyle T_{0}} 為材料常數。 SCG剪切模型 NP剪切模型 剪切鬆弛模量 參見 固體力學 流體力學 連續介質力學更多資訊 , ... 換算公式 均質各向同性線彈性材料具有獨特的彈性性質，因此知道彈性模量中的任意兩種，就可由下列換算公式求出其他所有的彈性模量。 ( 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 + 2 G 3 {\displaystyle \lambda +{\tfrac {2G}{3}}} E G 3 ( 3 G − E ) {\displaystyle {\tfrac {EG}{3(3G-E)}}} λ ( 1 + ν ) 3 ν {\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}} 2 G ( 1 + ν ) 3 ( 1 − 2 ν ) {\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}} E 3 ( 1 − 2 ν ) {\displaystyle {\tfrac {E}{3(1-2\nu )}}} M − 4 G 3 {\displaystyle M-{\tfrac {4G}{3}}} E = {\displaystyle E=\,} G ( 3 λ + 2 G ) λ + G {\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}} 9 K ( K − λ ) 3 K − λ {\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}} 9 K G 3 K + G {\displaystyle {\tfrac {9KG}{3K+G}}} λ ( 1 + ν ) ( 1 − 2 ν ) ν {\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}} 2 G ( 1 + ν ) {\displaystyle 2G(1+\nu )\,} 3 K ( 1 − 2 ν ) {\displaystyle 3K(1-2\nu )\,} G ( 3 M − 4 G ) M − G {\displaystyle {\tfrac {G(3M-4G)}{M-G}}} λ = {\displaystyle \lambda =\,} G ( E − 2 G ) 3 G − E {\displaystyle {\tfrac {G(E-2G)}{3G-E}}} K − 2 G 3 {\displaystyle K-{\tfrac {2G}{3}}} 2 G ν 1 − 2 ν {\displaystyle {\tfrac {2G\nu }{1-2\nu }}} E ν ( 1 + ν ) ( 1 − 2 ν ) {\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}} 3 K ν 1 + ν {\displaystyle {\tfrac {3K\nu }{1+\nu }}} 3 K ( 3 K − E ) 9 K − E {\displaystyle {\tfrac {3K(3K-E)}{9K-E}}} M − 2 G {\displaystyle M-2G\,} G = {\displaystyle G=\,} 3 ( K − λ ) 2 {\displaystyle {\tfrac {3(K-\lambda )}{2}}} λ ( 1 − 2 ν ) 2 ν {\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}} E 2 ( 1 + ν ) {\displaystyle {\tfrac {E}{2(1+\nu )}}} 3 K ( 1 − 2 ν ) 2 ( 1 + ν ) {\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}} 3 K E 9 K − E {\displaystyle {\tfrac {3KE}{9K-E}}} ν = {\displaystyle \nu =\,} λ 2 ( λ + G ) {\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}} E 2 G − 1 {\displaystyle {\tfrac {E}{2G}}-1} λ 3 K − λ {\displaystyle {\tfrac {\lambda }{3K-\lambda }}} 3 K − 2 G 2 ( 3 K + G ) {\displaystyle {\tfrac {3K-2G}{2(3K+G)}}} 3 K − E 6 K {\displaystyle {\tfrac {3K-E}{6K}}} M − 2 G 2 M − 2 G {\displaystyle {\tfrac {M-2G}{2M-2G}}} M = {\displaystyle M=\,} λ + 2 G {\displaystyle \lambda +2G\,} G ( 4 G − E ) 3 G − E {\displaystyle {\tfrac {G(4G-E)}{3G-E}}} 3 K − 2 λ {\displaystyle 3K-2\lambda \,} K + 4 G 3 {\displaystyle K+{\tfrac {4G}{3}}} λ ( 1 − ν ) ν {\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}} 2 G ( 1 − ν ) 1 − 2 ν {\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}} E ( 1 − ν ) ( 1 + ν ) ( 1 − 2 ν ) {\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}} 3 K ( 1 − ν ) 1 + ν {\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}} 3 K ( 3 K + E ) 9 K − E {\displaystyle {\tfrac {3K(3K+E)}{9K-E}}} 關閉 [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 （頁面存檔備份，存於互聯網檔案館）, 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. （原始內容存檔 (PDF)於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. （原始內容存檔於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. Wikiwand - on Seamless Wikipedia browsing. On steroids.
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