Vol. 9, No. 4, 2021

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Kelvin decomposition for nonlinear hyperelastic modeling in large deformation

Annie Morch, Jean-François Witz, Boris Desmorat, Rodrigue Desmorat and Mathias Brieu

Vol. 9 (2021), No. 4, 337–365
DOI: 10.2140/memocs.2021.9.337
Abstract

We propose using the Kelvin decomposition as a deformation projection tool to extend the linear elasticity formalism with fourth-order decomposition tensors in order to model nonlinear anisotropic hyperelastic behaviors in large deformation. We show how this decomposition makes it possible to generalize the Saint Venant–Kirchhoff model to a subclass of anisotropy. We also present a strategy to extend Ogden’s model to anisotropy. In the traditional Ogden approach, the eigenvalues of the strain tensor are used. We propose combining the Ogden model with the Kelvin decomposition in order to consider structural and stress-induced anisotropy.

An application is provided where the model parameters are optimized to fit both models on the experimental mechanical behavior of a textile reinforced elastomer. Results showed good accuracy between the experimental and modeled stress response.

Fourth-order projectors and the mathematical canvas make the analytical expression of the tangent elasticity tensor simpler. This method opens perspectives for easy implementation and modeling of linear and nonlinear anisotropic materials in finite element code.

Keywords
elasticity, Kelvin decomposition, hyperelasticity, anisotropy, large deformation
Mathematical Subject Classification
Primary: 74-05, 74-10, 74B20, 74E10
Milestones
Received: 11 February 2021
Revised: 10 November 2021
Accepted: 4 January 2022
Published: 14 March 2022

Communicated by Francesco dell'Isola
Authors
Annie Morch
Université de Lille, CNRS, UMR 9013 - LaMcube
Laboratoire de Mécanique Multiphysice Multiéchelle Villeneuve d’Ascq
France
Jean-François Witz
Université de Lille, CNRS, UMR 9013 - LaMcube
Laboratoire de Mécanique Multiphysice Multiéchelle Villeneuve d’Ascq
France
Boris Desmorat
Institut d’Alembert
Sorbonne Université
Paris
France
Rodrigue Desmorat
ENS Paris-Saclay, CNRS, LMT
Laboratoire de Mécanique et Technologie, Université Paris-Saclay
Gif-sur-Yvette
France
Mathias Brieu
Engineering, Computer Science, and Technology Mechanical Engineering
California State University
Los Angeles, CA
United States