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Pure Complementary Energy Principle and Analytical Solutions to General 3-D Nonlinear Elasticity

David Gao (Univ Ballarat/ANU)

SES Medal Symposium in honor of D.J. Steigmann

Mon 4:20 - 5:40

MacMillan 115

The speaker will present a detailed study on analytical solutions to a general nonlinear boundary-value problem in finite deformation theory. Based on the canonical duality theory and the pure complementary energy principle in nonlinear elasticity proposed by Gao in 1999 [1], we show that this type of nonlinear partial differential equations is actually equivalent to a coupled cubic algebraic (tensor) equation. For St Venant-Kirchhoff materials, this algebraic equation can be solved principally to obtain all possible solutions. Our results show that for any given statically admissible external source field, the problem has at least one global minimal solution in certain stable domain and at most eight solutions in the complementary domain. Additionally, the problem could have some unstable solutions. Applications will be illustrated by a complete set of analytical solutions for one-dimensional phase transitions problem of Ericksen’s bar. Results show that the global optimal solutions are usually nonsmooth, and cannot be captured by any traditional Newton-type direct approaches. This talk will demonstrate that the pure complementary energy principle and the triality theory play fundamental roles in nonconvex analysis and finite deformation theory. References: [1] Gao, David Y., General Analytic Solutions and Complementary Variational Principles for Large Deformation Nonsmooth Mechanics in Meccanica 34, 1999, 169-198 [2] Gao, D.Y.and Ogden, R.W. (2008) Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation, Quarterly J. Mech. Appl. Math. . 61 (4), 497-522