A short note on minor and major symmetries in linear elasticity

Simply applying the directional derivative either twice to the strain-energy density function (hyperelasticity) or once to the stress–strain state (Cauchy elasticity) does not lead to the symmetries of the fourth-order elasticity tensor specified in the literature. Moreover, there are many justifications and arguments for the desired symmetries, which are summarized in this contribution. Thus, a symmetrization operator has to be introduced to guarantee minor symmetry, since the symmetry of the strain tensor is frequently neglected but is needed to obtain results required for particular elasticity relations. A thorough investigation is provided for both Cauchy elasticity and hyperelasticity, and what conclusion can be drawn on by various assumptions.

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Elasticity tensor | Linear elasticity | Major symmetry | Minor symmetry

DDC Class

530: Physics

620.1: Engineering Mechanics and Materials Science

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https://creativecommons.org/licenses/by/4.0/

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s00419-025-02879-4.pdf

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A short note on minor and major symmetries in linear elasticity