Abstract
The aim of the paper is to characterize transformations that preserve the potential structure of a relationship between dual variables. The first step consists in deriving a geometric definition of the condition for the existence of a potential. Having at hand this formulation, it becomes clear that the canonical similitudes represents the class of transformations that preserves the potential form of a relationship. Next, we derive the conditions under which canonical similitudes preserve the convexity of the potential or change it into concavity. This new class of transformations can be viewed as a generalization of the Legendre-Fenchel transformation. These concepts are applied to the Hooke constitutive relationship.
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Vallée, C., Hjiaj, M., Fortuné, D. et al. Canonical and Anti-Canonical Transformations Preserving Convexity of Potentials. J Elast 103, 247–267 (2011). https://doi.org/10.1007/s10659-010-9283-4
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DOI: https://doi.org/10.1007/s10659-010-9283-4