The purpose of this paper is to establish a method of obtaining closed-form solutions in isotropic hyperelasticity using the complementary energy, the Legendre transform of the strain energy function. Using the complementary energy, the stress becomes the independent variable and the strain the dependent variable. Some of the implications of this formulation of the equations are explored and illustrative examples of solutions for spherical and cylindrical inflation for several forms of the complementary energy are presented.
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