Objective Tensor Rates and Applications in Formulation of Hyperelastic Relations |
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Authors: | S N Korobeynikov |
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Institution: | (1) Lavrentyev Institute of Hydrodynamics, Novosibirsk, 630090, Russia |
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Abstract: | Following Ogden, a class of objective (Lagrangian and Eulerian) tensors is identified among the second-rank tensors characterizing
continuum deformation, but a more general definition of objectivity than that used by Ogden is introduced. Time rates of tensors
are determined using convective rates. Sufficient conditions of objectivity are obtained for convective rates of objective
tensors. Objective convective rates of strain tensors are used to introduce pairs of symmetric stress and strain tensors conjugate
in a generalized sense. The classical definitions of conjugate Lagrangian (after Hill) and Eulerian (after Xiao et al.) stress and strain tensors are particular cases of the definition of conjugacy of stress and strain tensors in the generalized
sense used in the present paper. Pairs of objective stress and strain tensors conjugate in the generalized sense are used
to formulate constitutive relations for a hyperelastic medium. A family of objective generalized strain tensors is introduced,
which is broader than Hill’s family of strain tensors. The basic forms of the hyperelastic constitutive relations are obtained
with the aid of pairs of Lagrangian stress and strain tensors conjugate after Hill (the strain tensors in these pairs belong
to the family of generalized strain tensors). A method is presented for generating reduced forms of the constitutive relations
with the aid of pairs of Lagrangian and Eulerian stress and strain tensors conjugate in the generalized sense which are obtained
from pairs of Lagrangian tensors conjugate after Hill by mapping tensor fields on one configuration of a deformable body to
tensor fields on another configuration.
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Keywords: | Hyperelasticity Objective tensors Convective tensor rates |
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