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Multi-layered folding with voids

T. J. Dodwell, G. W. Hunt, M. A. Peletier and C. J. Budd
Philosophical Transactions: Mathematical, Physical and Engineering Sciences
Vol. 370, No. 1965, Geometry and mechanics of layered structures and materials (28 April 2012), pp. 1740-1758
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/23250117
Page Count: 19
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Multi-layered folding with voids
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Abstract

In the deformation of layered materials such as geological strata, or stacks of paper, mechanical properties compete with the geometry of layering. Smooth, rounded corners lead to voids between the layers, while close packing of the layers results in geometrically induced curvature singularities. When voids are penalized by external pressure, the system is forced to trade off these competing effects, leading to sometimes striking periodic patterns. In this paper, we construct a simple model of geometrically nonlinear multi-layered structures under axial loading and pressure confinement, with noninterpenetration conditions separating the layers. Energy minimizers are characterized as solutions of a set of fourth-order nonlinear differential equations with contact-force Lagrange multipliers, or equivalently of a fourth-order free-boundary problem. We numerically investigate the solutions of this free-boundary problem and compare them with the periodic solutions observed experimentally.

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