Pure Complementary Energy Principle and Triality Theory in Finite Elasticity

It is well-known that the complementary energy principle for large deformation elasticity was first proposed by Hellinger in 1914. Since Reissner clarified the boundary conditions in 1953, the complementary energy principles and methods in finite deformation mechanics have been studied extensively during the last forty years (cf. e.g. Koiter, 1976; Nemat-Nasser, 1977; Atluri, 1980; Lee & Shield, 1980; Buffer, 1983; Oden & Reddy, 1983; Ogden, 1984; Tabarrok, 1984 and much more). But the Hellinger-Reissner principle involves both the second Piola-Kirchhoff stress and the displacement, it is not considered as a pure complementary energy principle. For more than 80 years, this principle was considered only as a stationary principle. Its extremum property has been an open problem, which yielded many arguments. The Levinson-Zubov principle involves only the first Piola-Kirchhoff stress r . Unfortunately in finite deformation problems with compressive external loads, the stored energy function W(F) is usually nonconvex in the deformation gradient F (cf. e.g. Ogden, 1984). In this case, even for one-dimensional problems, the complementary energy W e obtained by the Legendre transformation