Abstract
The use of the Doyle-Ericksen formula in the context of the spatial description in elasticity is discussed and critically examined. Motivated by the observation that the aforementioned formula is not 3uitable for a spatial description, a dual picture of elasticity is derived according to which the metric at the reference configuration is playing a central role. Hereby the free energy is considered to be a direct function of the Almansi strain tensor (or alternatively the inverse left Cauchy-Green tensor) leading to relations well suited for a spatial description. A new stress tensor to be regarded as the dual variable of the Almansi strain tensor is defined. Its geometric interpretation leads further to the statement that the rotated (isometric) Cauchy stress tensor can be derived by varying the free energy function with respect to the metric of the reference configuration. Moreover, by considering the isometric tensors of the aforementioned strain tensors we arrive at the conclusion that the Cauchy stress tensor can be derived by varying the free energy function with respect to the inverse of the metric at the actual configuration. These statements can be considered as the counterpart of the Doyle-Ericksen formula in a spatial description.
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