  The main difference between 3D and 4D printed structures is one extra
dimension that is smart evolution over time. However, currently, there is no
general formula to model and predict this extra dimension. Here, by starting
from fundamental concepts, we derive and validate a universal bi-exponential
formula that is required to model and predict the fourth D of 4D printed
multi-material structures. 4D printing is a new manufacturing paradigm to
elaborate stimuli-responsive materials in multi-material structures for
advanced manufacturing (and construction) of advanced products (and
structures). It conserves the general attributes of 3D printing (such as the
elimination of molds, dies, and machining) and further enables the fourth
dimension of products and structures to provide intelligent behavior over time.
This intelligent behavior is encoded (usually by an inverse mathematical
problem) into stimuli-responsive multi-materials during printing and is enabled
by stimuli after printing. Here, we delve into the fourth dimension and reveal
three general laws that govern the time-dependent shape-shifting behaviors of
almost all (photochemical-, photothermal-, solvent-, pH-, moisture-,
electrochemical-, electrothermal-, ultrasound-, enzyme-, etc.-responsive)
multi-material 4D structures. We demonstrate that two different types of
time-constants govern the shape-shifting behavior of almost all the
multi-material 4D printed structures over time. Our results starting from the
most fundamental concepts and ending with governing equations can serve as
general design principles for future research in the 4D printing field, where
the time-dependent behaviors should be understood, modeled, and predicted,
correctly. Future software and hardware developments in 4D printing can also
benefit from these results.

Applied Physics