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A study on hyperelastic models for micromorphic solids

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Abstract

This paper presents two hyperelastic models for the micromorphic hyperelasticity which can be efficiently utilized for materials with high dependence on the microdeformation gradient. To this end, two new microdeformation gradient-based strain measures are introduced and used in hyperelastic formulation. The developed formulation for the micromorphic hyperelasticity makes it possible to define hyperelastic functions whose dependency on the microdeformation gradient can be clearly discussed. Also, based on the proposed formulation, any kind of hyperelastic models can be formulated using the defined strain measures.

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Data Availability Statement

Data sharing is not applicable to this article as it describes entirely theoretical research.

Abbreviations

\(\nabla_{x} ,\nabla_{X}\) :

Del operator in current and reference coordinate systems

\(\mathbf{x},\mathbf{X}\) :

Macroelement position vector in current and reference configurations

\(\widetilde{\mathbf{x}},\widetilde{\mathbf{X}}\) :

Microelement position vector in current and reference configurations

\({\varvec{\upxi}},{\varvec{\Xi}}\) :

Microelement position vector with respect to macroelement in current and reference configurations

\(\mathbf{F},{\varvec{\upchi}}\) :

Deformation gradient tensor and microdeformation tensor

\(\mathbf{C}\) :

Symmetric second-rank strain measure tensor known as the right Cauchy–Green deformation tensor

\({\varvec{\Upsilon}}\) :

Symmetric second-rank strain measure tensors indicating microdeformation

\(\widehat{{\varvec{\Gamma}}},{\varvec{\Gamma}}\) :

Two third-rank strain measure tensors

\({\varvec{\Lambda}},\mathbf{K}\) :

New second-rank strain measure tensor

\(\widetilde{\mathbf{t}},\mathbf{n}\) :

Traction vector and its normal vector

\({\varvec{\upsigma}},\mathbf{s},{\varvec{\upmu}}\) :

The Cauchy stress tensor, second-rank stress tensor and stress moment tensor

\( {\mathbf{t}} = {\varvec{\upsigma}} - {\mathbf{s}} \) :

Microstress tensor

\( \psi \) :

Helmholtz free energy

\( \prod \) :

Potential energy

\(I_{1}^{{\mathbf{A}}} ,I_{2}^{{\mathbf{A}}} ,I_{3}^{{\mathbf{A}}}\) :

Principal invariants of any second-rank tensor \(\mathbf{A}\)

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  1. Faculty of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

    M. Bazdid-Vahdati, R. Ansari & A. Darvizeh

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  1. M. Bazdid-Vahdati
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Correspondence to R. Ansari.

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Bazdid-Vahdati, M., Ansari, R. & Darvizeh, A. A study on hyperelastic models for micromorphic solids. Eur. Phys. J. Plus 138, 87 (2023). https://doi.org/10.1140/epjp/s13360-022-03637-z

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