Nonuniform, nonlinear deformations of four-parallelogram origami via a symmetry-based continuum theory

Origami sheets, particularly those with parallelogram faces, are archetypal examples of flexible mechanical metamaterials. However, while they possess well-defined uniform deformation modes, generic loading conditions lead to spatially complex patterns of strain and curvature, raising the question of which low-energy deformations are permitted. In this talk, we present a continuum theoretical approach to characterizing the nonuniform, nonlinear deformations of such origami sheets. We identify a mechanism field which describes the spatially varying activation of the sheet's intrinsic rigid mode. By exploiting various symmetries, we are able to identify a new nonlinear compatibility condition, a scalar partial differential equation, that governs this field's activation. We verify the validity of our approach using origami simulation software (MERLIN) over a wide range of embedding geometries and loads, with load amplitudes ranging from linear to nonlinear. This approach sheds new light on how origami and related structures that combine flexibility, strength and curvature can yield rich and varied spatial structures.