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

• Published in: Journal of engineering mathematics Vol. 103; no. 1; pp. 127 - 157
• Main Authors: dell’Isola, Francesco, Cuomo, Massimo, Greco, Leopoldo, Della Corte, Alessandro
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.04.2017; Springer Nature B.V; Springer Verlag
• Subjects: Applications of Mathematics; Bias; Computational Mathematics and Numerical Analysis; Fibers; Internal energy; Mathematical and Computational Engineering; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Mechanics; Mechanics of materials; Physics; Theoretical and Applied Mechanics; 74A60; Inextensible fibre network; Second gradient theory; Bias extension test; Pantographic sheet; 74B20; 74K15; 74A30; Constrained optimization
• ISSN: 0022-0833; 1573-2703
• DOI: 10.1007/s10665-016-9865-7

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.