Wrinkling of soft composite sheets

We examine the buckling shape and critical compression of a confined composite sheet floating on a liquid. The sheet is inhomogeneous due to the presence of liquid inclusions, such that its elastic properties vary in the direction of uniaxial compression. From the Föppl–von Kármán model of Hookean elasticity, we formulate a fourth order eigenvalue problem where the eigenvalue is the compressive load imposed on the sheet and the eigenfunction is the midplane displacement. Assuming that the volume fraction of the liquid inclusions is small, we can solve the eigenvalue problem perturbatively. To leading order the sheet is homogeneous and we find that the buckling shape has a symmetric and an antisymmetric buckling mode. The mode associated with the minimum compressive load depends on the natural length of the compressed sheet. The symmetry changes are realized at the next order of the analysis, where the liquid inclusions are influential. Hence, we treat the inhomogeneities using a Galerkin method with enforced P-T symmetry. In the case of a large sheet, we show that the wrinkles are localized and are treated using a WKB method.