**1. Introduction**

In order to survive in variable environments and keep their performance, natural materials have evolved to be active and adaptive, retaining functions across a range of stresses or strains or changing thermomechanical properties in response to external stimuli like light, temperature or moisture [1]. Several active materials have been developed to mimic the unique properties of natural materials and adaptive structures [2]. Among these active materials, shape-memory polymers (SMPs) have attracted much attention due to their lower density, higher recoverable strain of up to 400%, lower cost, simple shape programming procedure, and excellent controllability over the recovery temperature [3,4].

In the recent two decades, three-dimensional (3D) printing technology, also known as additive manufacturing (AM), has gained considerable attention as an advanced manufacturing technique that can create complex objects through depositing materials in a layer-by-layer manner [5–9]. With the introduction of active materials, 3D printing approaches have shown excellent potential for the fabrication of adaptive structures, namely four-dimensional (4D) printed structures, with the capability of reshaping their configuration and changing their properties over time [10–12]. For the first time, Tibbits [13] experimentally demonstrated how 4D-printed objects could transform over time and perform self-assemblies. While 3D printing methods can be used to fabricate static structures, 4D printing methods allow the fabrication of dynamically reconfigurable architectures with desired functionality and responsiveness. Considering a specific application, 4D-printed objects can be designed to respond to environmental stimuli and external triggers such as humidity, light, heat, and electric or magnetic fields [14–16]. For example, by 4D printing temperature-sensitive SMPs, Bodaghi et al. [17] introduced adaptive metamaterials with the capability of 1D/2D-to-2D/3D shape-shifting through self-folding and/or self-coiling. Wang et al. [18] introduced a novel 4D printing technology of composites with continuous embedded fibers for which the programmable deformation was caused by the difference in coefficient of thermal expansion between continuous fibers and flexible matrix. Zhang et al. [19] 4D-printed lightweight structures with self-folding/unfolding performances when exposed to a certain temperature. Also, Zhao et al. [20] 4D-printed SMPs by stereolithography of photopolymers. Fold-deploy test and shape-memory cycle measurements proved a high shape recovery rate, shape fixity, and recovery of the printed objects. Zolfagharian et al. [21] achieved controlled bending in a commercial prestrained SMP film by using different 4D-printed patterns and a number of layers. The photothermal stimulus was used to induce differential shrinking through the thickness of the actuator hinge. Recently, Wu et al. [10] designed a new 4D printable acrylate-based photosensitive resin prepared for digital light processing. They also investigated the influence of crosslinker concentration on the shape memory and mechanical properties.

Acoustic metamaterials are a new class of architected materials designed to control, direct, and manipulate waves through the material. Wave propagation in periodic metamaterials has been investigated broadly. Research works have revealed that these architected structures can be designed to gain desirable dispersion performance [22], lightweight structures [23], and excellent energy absorbers [24] by arranging unit cells in periodic repetition. The dispersion performance may present stop-band and pass-band frequency ranges as well. Phani et al. [25] introduced a computational procedure to find bandgaps by using Floquet–Bloch principles. Matlack et al. [26] fabricated metastructures by combining 3D printing polycarbonate lattices with steel cubes assembling. They showed that the value of bandgap could be varied by changing the lattice geometry and structural stiffness. By using deployable structures composed of beams and torsion springs, Nadda and Karami [27] showed that wave transmission characteristics could be tuned by modulating the fold angle. By combining a periodic lattice and locally-resonant inclusions with different temperature dependency, Nimmagadda and Matlack [28] proposed a metastructure that can tune the bandgaps under thermal stimuli.

The main objective of this research is to show how 4D printing technology can be used to engineer adaptive metastructures with the ability to control elastic wave propagation. Such structure is essential in vibration mitigation and acoustic attenuation. Inspired by thermomechanics of SMPs and the potential of fused deposition modeling (FDM) in 4D printing self-bending elements, adaptive functionally graded (FG) beams are fabricated. It is shown, experimentally and numerically, how 4D printing speed can control shape recovery and self-bending features of active elements. A 3D constitutive model, along with an in-house finite element (FE) code programmed in MATLAB software, is developed to replicate shape recovery of FG active elements 4D-printed at four different printing speeds. Afterward, a simple approach, along with the commercial FE software package of COMSOL Multiphysics, is introduced to simulate the material tailoring in the 4D printing stage and self-bending of active SMP elements. The accuracy and simplicity of the straightforward FE approach implemented

in commercial FE software are assured via comparative studies with experimental data and numerical results from the in-house FE MATLAB-based code. Two periodic architected temperature-sensitive metastructures with adaptive dynamical characteristics are conceptually proposed. Their dynamic behaviors during heating–cooling processes are numerically studied in detail. Adaptive wave propagation (pass-band and stop-band), desired dynamic performance, and vibration manipulation are numerically demonstrated to be some of the unique characterizations of these metastructures. Computational studies reveal that, while the bandgaps in these metastructures are induced by the self-bending mechanism, their crucial feature is that the bandgap size and frequency range can be controlled and broadened through local resonances linked to changes in the structural geometry. The material/structural formulation, concepts, and results provided in this article are expected to be instrumental towards 4D printing adaptive metastructures for a broad range of applications, including structural vibration absorption, waveguiding, and noise mitigation.
