JElast https://doi.org/10.1007/s10659-018-9677-2 Finite Third-order Gradient Elasticity and Thermoelasticity 1 1 Jörg Christian Reiher · Albrecht Bertram Received: 12 November 2017 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract A constitutive format for the third-order gradient elasticity is suggested. It in- cludes both isotropic and anisotropic non-linear behavior under ﬁnite deformations. Appro- priate invariant stress and strain variables are introduced, which allow for reduced forms of the elastic energy law that identically fulﬁll the objectivity requirement. After working out the transformation behavior under a change of the reference placement, the symmetry transformations for third-order materials can be introduced. After the mechanical third-order theory, an extension to thermoelasticity is given, and necessary and sufﬁcient conditions are derived from the Clausius-Duhem inequality. Keywords Third-order elasticity · Second strain gradient elasticity · Finite gradient elasticity Mathematics Subject Classiﬁcation (2000) 74A99 · 74B20 1 Introduction After the seminal papers of Toupin  and Mindlin  on gradient materials, such exten- sions of the classical simple materials have become the subject of steadily growing research interest. Mostly surface effects are addressed as in [12, 13, 28, 34, 39], dislocation phenom- ena as in [16, 25, 26], or length scale effects as in
Journal of Elasticity – Springer Journals
Published: May 29, 2018
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