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Solving hyperelastic material problems by asymptotic numerical method

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Abstract

This paper presents a numerical algorithm based on a perturbation technique named asymptotic numerical method (ANM) to solve nonlinear problems with hyperelastic constitutive behaviors. The main advantages of this technique compared to Newton–Raphson are: (a) a large reduction of the number of tangent matrix decompositions; (b) in presence of instabilities or limit points no special treatment such as arc-length algorithms is necessary. The ANM uses high order series approximation with auto-adaptive step length and without need of any iteration. Introduction of this expansion into the set of nonlinear equations results into a sequence of linear problems with the same linear operator. The present work aims at providing algorithms for applying the ANM to the special case of compressible and incompressible hyperelastic materials. The efficiency and accuracy of the method are examined by comparing this algorithm with Newton–Raphson method for problems involving hyperelastic structures with large strains and instabilities.

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Authors and Affiliations

  1. Laboratoire de Physique et Mécanique des Matériaux, FRE CNRS 3236, Université Paul Verlaine Metz, Ile du Saulcy, 57045, Metz Cedex 01, France

    Saeid Nezamabadi & Hamid Zahrouni

  2. Laboratoire Modélisation et Simulation Multi Echelle, UMR CNRS 8208, Université Paris-Est, 5 Bd Descartes, 77454, Marne-la-Vallée Cedex 02, France

    Julien Yvonnet

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  1. Saeid Nezamabadi
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  2. Hamid Zahrouni
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  3. Julien Yvonnet
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Correspondence to Saeid Nezamabadi.

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Nezamabadi, S., Zahrouni, H. & Yvonnet, J. Solving hyperelastic material problems by asymptotic numerical method. Comput Mech 47, 77–92 (2011). https://doi.org/10.1007/s00466-010-0531-z

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