  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.
