Nonlinear Logarithmic Hyperelasticity with Isotropy in the Initial State
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2023-06
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Journal Article
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Abstract
Nonlinear stress–strain relations for hyperelasticity with isotropy in the initial state can be modeled by strain energies as functions of three (scalar) independent strain or deformation invariants. A model based on logarithmic strain tensors is compared to Rubin’s model based on unimodular Cauchy–Green deformation tensors. Using the three ordered eigenvalues determined by Cardano’s formula, three Mohr’s circles can be constructed for each of the stress and deformation/strain tensors involved. In Cardano’s formula and Mohr’s three circles, the spherical contributions resulting from the trace of the tensor and the deviatoric contributions resulting from the norm and determinant of its deviator are separated additively. On the one hand, when an elasticity model such as Rubin’s is based on (unimodular) Cauchy–Green deformation tensors, their traces or deviators have no simple direct interpretation, so that dilatation and distortion are coupled multiplicatively. The stress–strain relations then employ material functions of all three invariants, and Lode’s angles of the stress and deformation tensors generally differ. When, on the other hand, the elasticity model is based on logarithmic strain tensors, their traces and deviators do have the physical interpretation of finite dilatation and finite distortion. Dilatation and distortion are then decoupled additively. The question arises whether Lode’s angles of the stress and logarithmic strain should be different at all, and a simple isotropic elasticity model with matching Lode’s angles and the strain energy as a function of only two (dilatation and distortion) invariants is presented.
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published
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33 (3)
Pages / Article No.
47
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Springer
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Subject
Coaxial/collinear tensors; Isotropy in the initial state; Lode’s angle; Logarithmic strain; Mohr’s circles; Nonlinear hyperelasticity