Theoretical stiffness limits of 4D printed self-folding metamaterials

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. However, the opposite stiffness requirements for the folding process and the subsequent loadbearing of 3D structures impose an intrinsic limitation in designing self-folding metamaterials: while a low stiffness is required for the successful completion of the self-folding step, a high stiffness is needed for utilizing the folded structure as a load-bearing mechanical metamaterial. Here, we present a nonlinear analytical model of self-folding bilayer constructs composed of an active and passive layer. This finite-deformation theoretical model predicts the curvature of activated bilayers, establishes their stability limits, and estimates the stiffness of folded bilayers, yielding the theoretical stiffness limits of self-folding bilayers. We use our model to identify the optimal combinations of geometrical and mechanical properties that result in the highest possible stiffness of folded constructs. We then compare the predictions of our analytical model with computational results, and validate our theory with experimental realizations of 4D printed structures. 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.