Theoretical stiffness limits of 4D printed self-folding metamaterials

Abstract: 4D printing of flat sheets that self-fold into architected 3D structures is a powerful origami-inspired approach for the fabrication of multi-functional devices and metamaterials. The possibility to endow the initially flat sheet with a variety of surface-related functionalities provides the means to simultaneously achieve multi-functional performance and a complex 3D architecture. One intrinsic limitation of such a production strategy is the contradictory stiffness requirements for the folding and for subsequent load-bearing steps: while a low stiffness is required to allow for the successful completion of the intended self-folding process, a high stiffness is needed for the subsequent application of the folded structure as a load-bearing mechanical metamaterial. The competition between these two requirements determines the theoretical limits of 4D printed self-folding mechanical metamaterials, which have not been determined before. Here, we present a nonlinear analytical model of self-folding bilayer constructs composed of an active and a passive layer. This finite deformation theoretical model predicts the curvature of activated bilayers, establishes their stability limits, and estimates the stiffness of folded bilayers. Combining the three abovementioned aspects allows for establishing the theoretical stiffness limits of self-folding bilayers. The optimal combinations of geometrical and mechanical properties that result in the highest possible stiffness of folded constructs can then be determined. We compare the predictions of our analytical models with computational as well as experimental results. Finally, we evaluate the theoretical stiffness limits of bilayer constructs made using the most common types of stimuli-responsive materials. Our analysis shows that a maximum effective modulus of ~1.5 GPa can be achieved using the currently available shape-memory polymers.