Abstract
We consider a class of models motivated by previous numerical studies of wrinkling in highly stretched, thin rectangular elastomer sheets. The model is characterized by a finite-strain hyperelastic membrane energy perturbed by small bending energy. In the absence of the latter, the membrane energy density is not rank-one convex for general spatial deformations but reduces to a polyconvex function when restricted to planar deformations, i.e., two-dimensional hyperelasticity. In addition, it grows unbounded as the local area ratio approaches zero. The small bending component of the model is the same as that in the classical von Kármán model. The latter penalizes arbitrarily fine-scale wrinkling, resolving both the amplitude and wavelength of wrinkles. Here, we prove the existence of energy minima for a general class of such models.
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“Only a mathematical existence proof can ensure that the mathematical description of a physical phenomenon is meaningful”.
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Acknowledgements
This work was supported in part by the National Science Foundation through grant DMS-2006586, which is gratefully acknowledged. I thank Gokul Nair for useful comments on an earlier version of the work.
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TJH carried out the research and wrote the manuscript.
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Healey, T.J. An existence theorem for a class of wrinkling models for highly stretched elastic sheets. Z. Angew. Math. Phys. 74, 221 (2023). https://doi.org/10.1007/s00033-023-02111-9
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DOI: https://doi.org/10.1007/s00033-023-02111-9