Experimental results were corroborated with theoretical expectations by developing a numerical model of filament actuation, which is summarized here. Complete details are provided in the Supplementary Materials. In the model, the filament was treated as a thin elastic beam composed of 40 individual rigid segments. Because of the small size of each segment, each segment was treated as having a uniform magnetization and experiencing both a uniform magnetic field and a uniform magnetic field gradient, determined by its location relative to a simulated permanent magnet. The magnetization of the filament in the presence of a magnetic field gradient leads to an attractive magnetic force on each segment, favoring displacement toward the magnet. Elastic or gravitational forces may, however, restrict this motion. The net force on each segment of the filament due to magnetism, elasticity, and gravity (neglected for the snappers) determines the instantaneous acceleration of the filament in any configuration, from which its dynamics and equilibrium configurations may be determined.

Using an algorithm written in MATLAB R2017a, the field geometry in the neighborhood of the filament was first simulated, and then the net force on each segment of the filament was calculated. The net force was applied over a short time step to update the momentum of each segment, and the momentum was used to update the position of each segment over the same time interval. At each step, the momentum was reduced by a small fraction (0.1 to 0.5%) to dampen internal oscillations. This process was iterated until the average net force per segment fell below a minimum threshold value, which was set to 10 to 100 nN, depending on the system.

Elastic forces were modeled as two distinct components: bending and extension. Elastic bending forces were determined from Euler-Bernoulli beam theory under the assumption that each segment has a constant curvature. Elastic extension/compression forces always act parallel to the filament and are proportional to the modulus, cross-sectional area, and strain of each segment.

The magnetic field was taken to be that of an ideal cylindrical magnet and was modeled in the xy plane using an established algorithm (60). Each segment of the filament then attained a net volume magnetization according to the local field magnitude, where the magnetization curve of each of the materials, M(B), has been measured (fig. S1). Differentiating the magnetic energy of each segment results in the net magnetic force, which depends on the local magnetic field magnitude, local magnetic field gradient, and the susceptibility of the filament, which was taken from the magnetization curves.

Dimensions of each modeled filament were chosen to match experimental configurations. The elastic modulus of IROGRAN at 25°C and the temperature-dependent elastic modulus of DiAPLEX were obtained from tensile testing of samples loaded with Fe particles (fig. S3). For modeling the shape memory behavior of DiAPLEX, its modulus was divided into two parts: The permanent configuration was assumed to have a fixed modulus equal to the minimum of E(T) = E(80°C), while the temporary configuration had a temperature-dependent modulus comprising the remainder, or E(T) − E(80°C).

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