Abstract
The rari-constant theory of linear elasticity is based on the assumption that elasticity in solids is caused by only pair potentials with coaxial forces acting between atoms. The strain energy of each pair potential depends on the square of the strain between the atoms in the pair. This strain can be determined by taking the inner product of the strain tensor with a structural tensor that is the tensor product of a unit vector with itself. It is shown that the 15 independent constants in the rari-constant theory can be generated by a complete set of 15 structural tensors. It is also shown that the 6 additional independent constants in the multi-constant theory can be generated by taking the inner product of 6 of these structural tensors with the square of the strain tensor. A generalization of these results for nonlinear elasticity is discussed with reference to recent work which compares the structural and generalized structural tensor approaches to modeling fibrous tissues.
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Acknowledgements
This research was also partially supported by M.B. Rubin’s Gerard Swope Chair in Mechanics. M.B. Rubin would like to acknowledge J. Goddard for bring this issue to his attention and would also like to acknowledge the gracious hospitality of ETH in Zurich during part of his sabbatical leave from Technion. The authors would also like to acknowledge R.L. Fosdick for directing our attention to [6].
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Rubin, M.B., Ehret, A.E. Invariants for Rari- and Multi-Constant Theories with Generalization to Anisotropy in Biological Tissues. J Elast 133, 119–127 (2018). https://doi.org/10.1007/s10659-018-9674-5
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DOI: https://doi.org/10.1007/s10659-018-9674-5