Abstract
This paper addresses some fundamental issues in nonconvex analysis. By using pure complementary energy principle proposed by the author, a class of fully nonlinear partial differential equations in nonlinear elasticity is able to convert a unified algebraic equation, a complete set of analytical solutions are obtained in dual space for 3-D finite deformation problems governed by generalized neo-Hookean model . Both global and local extremal solutions to the nonconvex variational problem are identified by a triality theory. Connection between challenges in nonlinear analysis and NP-hard problems in computational science is revealed. Results show that Legendre–Hadamard condition can only guarantee ellipticity for generalized convex problems. For nonconvex systems, the ellipticity depends not only on the stored energy, but also on the external force field. Uniqueness is proved based on a generalized quasiconvexity and a generalized ellipticity condition. Application is illustrated for nonconvex logarithm stored energy.
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Notes
- 1.
It was proved recently that rank-one convexity also implies polyconvexity for isotropic, objective, and isochoric elastic energies in the two-dimensional case [15].
- 2.
The neighborhood \({\mathscr {X}}_o\) of \({\varvec{\chi }}_k\) in the canonical duality theory means that \({\varvec{\chi }}_k\) is the only one critical point of \(\varPi ({\varvec{\chi }})\) on \({\mathscr {X}}_o\) (see [5]).
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Acknowledgements
The research was supported by US Air Force Office of Scientific Research (AFOSR FA9550-10-1-0487).
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Gao, D.Y. (2017). Analytic Solutions to Large Deformation Problems Governed by Generalized Neo-Hookean Model. In: Gao, D., Latorre, V., Ruan, N. (eds) Canonical Duality Theory. Advances in Mechanics and Mathematics, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-319-58017-3_2
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