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Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies

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Abstract

We consider a bi-dimensional sheet consisting of two orthogonal families of inextensible fibres. Using the representation due to Rivlin and Pipkin for admissible placements, i.e. placements preserving the lengths of the inextensible fibres, we numerically simulate a standard bias extension test on the sheet, solving a non-linear constrained optimization problem. Several first and second gradient deformation energy models are considered, depending on the shear angle between the fibres and on its gradient, and the results obtained are compared. The proposed numerical simulations will be helpful in designing a systematic experimental campaign aimed at characterizing the internal energy for physical realizations of the ideal pantographic structure presented in this paper.

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Acknowledgments

The authors gratefully acknowledge the financial support of the International Research Center on the Mathematics and Mechanics of Complex Systems (M&MoCS). The interesting discussions with the participants of the workshop Computational Mechanics of Generalized Continua and Applications to Materials with Microstructure [Catania, 29–31 October 2015, Scuola Superiore di Catania, and the François Cosserat-Tullio Levi Civita International Laboratory associated to CNRS (LIA Coss&Vita)] were very fruitful and greatly influenced this work.

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Authors and Affiliations

  1. Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Rome, Italy

    Francesco dell’Isola

  2. Department of Civil and Environmental Engineering, University of Catania, Catania, Italy

    Massimo Cuomo

  3. International Research Center on the Mathematics and Mechanics of Complex Systems, University of L’Aquila, L’Aquila, Italy

    Leopoldo Greco

  4. Department of Mechanical and Aerospace Engineering, Sapienza University of Rome, Rome, Italy

    Alessandro Della Corte

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  1. Francesco dell’Isola
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  2. Massimo Cuomo
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  4. Alessandro Della Corte
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Correspondence to Alessandro Della Corte.

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dell’Isola, F., Cuomo, M., Greco, L. et al. Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies. J Eng Math 103, 127–157 (2017). https://doi.org/10.1007/s10665-016-9865-7

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