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非线性弹性理论变分原理的统一理论

郭仲衡

郭仲衡. 非线性弹性理论变分原理的统一理论[J]. 应用数学和力学, 1980, 1(1): 5-23.
引用本文: 郭仲衡. 非线性弹性理论变分原理的统一理论[J]. 应用数学和力学, 1980, 1(1): 5-23.
Guo Zhong-heng. Unified Theory of Variational Principles in Non-linear Theory of Elasticity[J]. Applied Mathematics and Mechanics, 1980, 1(1): 5-23.
Citation: Guo Zhong-heng. Unified Theory of Variational Principles in Non-linear Theory of Elasticity[J]. Applied Mathematics and Mechanics, 1980, 1(1): 5-23.

非线性弹性理论变分原理的统一理论

Unified Theory of Variational Principles in Non-linear Theory of Elasticity

  • 摘要: 本文旨在介绍和讨论非线性弹性理论的几个主要变分原理——古典的势能原理,余能原理以及目前争论甚多的另两个余能原理(Levinson原理和Fraeijs de Veubeke原理),同时给出了和这些原理相对应的广义变分原理.本文单一地从虚功原理出发,系统推导并严格论证了这些原理,并且指出了各原理间的内在联系.出发点是一个,采取不同的变量和Legendre变换就导致不同的原理.这样,各变分原理在统一的框架里构成一个有机的整体.文中未涉及的其它原理也同样可以纳入这框架.给出的关系图使读者更能看清各原理间的纵横关系.
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    [9] Zubov,L.M.,The stationary principle of complementary work in nonlinear theory of elasticity,Prikl,Mat.Meh.,34(1970),228-232.
    [10] Koiter,W.T.,On the principle of stationary complementary energy in the nonlinear theory of elasticity,SIAM,J.Appl.Math.,25(1973),424-434.
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    [13] Christoffersen,J.,On Zubov's principle of stationary complementary ebergy and a related principle,Rep.No.44,Danish Center for Appl.Math,and Mech.,April,(1973).
    [14] Ogden,R.W.,A note on variational theorems in non-linear elastostatics,Math.Proc.Cams.Phil.Soc.,77(1975),609-615.
    [15] Dill,E.H.,The complementary energy principle in nonlinear elasticity,Lett.Appl.and Engng.Sci,5(1977),95-106.
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    [17] Ogden.R.W.,Extremum principles in non-linear elasticity and their application to composite-Ⅰ,Int.J.Solids Struct.,14(1978),265-282.
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    [20] Macvean,D.B.,Die Elementararbeit in einem Kontinuum und die Zuordnung von Spannungs-und Verzerrungstensoren,ZAMP,19(1968),157-185.
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    [24] 钱伟长,弹性理论中广义变分原理的研究及共在有限元计算中的应用,清华大学科学报告,1978年11月.
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出版历程
  • 收稿日期:  1979-07-30
  • 刊出日期:  1980-02-15

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