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Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Geometric nonlinearity: potential energy, complementary energy, and the gap function


Authors: Yang Gao and Gilbert Strang
Journal: Quart. Appl. Math. 47 (1989), 487-504
MSC: Primary 73C50; Secondary 49H05, 73B99, 73G05
DOI: https://doi.org/10.1090/qam/1012271
MathSciNet review: MR1012271
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Abstract | References | Similar Articles | Additional Information

Abstract: Dual minimum principles for displacements and stresses are well established for linear variational problems and also for nonlinear (and monotone) constitutive laws. This paper studies the problem of geometric nonlinearity. By introducing a gap function, we recover complementary variational principles in the equilibrium problems of mathematical physics. When the gap function is nonnegative those become minimum principles. The theory is based on convex analysis, and the applications made here are to nonlinear mechanics.


References [Enhancements On Off] (What's this?)

  • Gilbert Strang, Introduction to applied mathematics, Wellesley-Cambridge Press, Wellesley, MA, 1986. MR 870634
  • Ivar Ekeland and Roger Temam, Analyse convexe et problèmes variationnels, Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). Collection Études Mathématiques. MR 0463993
  • P. D. Panagiotopoulos, Inequality problems in mechanics and applications, Birkhäuser Boston, Inc., Boston, MA, 1985. Convex and nonconvex energy functions. MR 896909
  • E. Reissner, Variational principles in elasticity, Finite Element Handbook, H. Kardestuncer, ed., McGraw-Hill, New York, 1987 E. Tonti, A mathematical model for physical theories, Fisica Matematica, Serie VIII, LII, 1972 Yang Gao and Gilbert Strang, Dual extremum principles in finite deformation elastoplastic analysis, to be published
  • Yang Gao and Tomasz Wierzbicki, Bounding theorem in finite plasticity with hardening effect, Quart. Appl. Math. 47 (1989), no. 3, 395–403. MR 1012265, DOI https://doi.org/10.1090/qam/1012265
  • Włodzimierz Robert Bielski and Józef Joachim Telega, On the complementary energy principle in finite elasticity, Proceedings of the International Conference on Nonlinear Mechanics (Shanghai, 1985) Sci. Press Beijing, Beijing, 1985, pp. 211–218. MR 1037274
  • W. R. Bielski and J. J. Telega, The complementary energy principle in finite elastostatics as a dual problem, Finite rotations in structural mechanics (Jabłonna, 1985) Lecture Notes in Engrg., vol. 19, Springer, Berlin, 1986, pp. 62–81. MR 870033, DOI https://doi.org/10.1007/978-3-642-82838-6_5
  • A. Hanyga, Mathematical Theory of Non-Linear Elasticity, Horwood-John Wiley, New York, 1985

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Article copyright: © Copyright 1989 American Mathematical Society