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: 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.
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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
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A. Hanyga, Mathematical Theory of Non-Linear Elasticity, Horwood-John Wiley, New York, 1985
Gilbert Strang, Introduction to Applied Mathematics, Wellesley-Cambridge Press, Wellesley, MA, 1986
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976
P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications, Birkhäuser, Basel, 1985
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 T. Wierzbicki, Bounding theorem in finite plasticity with hardening effect, Quart. Appl. Math. 47, 395–403 (1989)
J. J. Telega and W. R. Bielski, On the complementary energy principle in finite elasticity, Intern. Conf. on Nonlinear Mechanics, Science Press, Beijing, 1985
J. J. Telega and W. R. Bielski, The complementary energy principle in finite elastostatics as a dual problem, Lecture Notes in Engineering 19, 62–81 (1986)
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