Analytical modeling of a magnetoactive elastomer unimorph

Magnetoactive elastomers (MAEs) are capable of large deformation, shape programming, and moderately large actuation forces when driven by an external magnetic field. These capabilities enable applications such as soft grippers, biomedical devices, and actuators. To facilitate complex shape deformation and enhanced range of motion, a unimorph can be designed with varying geometries, behave spatially varying multi-material properties, and be actuated with a non-uniform external magnetic field. To predict actuation performance under these complex conditions, an analytical model of a segmented MAE unimorph is developed based on beam theory with large deformation. The effect of the spatially-varying magnetic field is approximated using a segment-wise effective torque. The model accommodates spatially varying concentrations of magnetic particles and differentiates between the actuation mechanisms of hard and soft magnetic particles by accommodating different assumptions concerning the magnitude and direction of induced magnetization under a magnetic field. To validate the accuracy of the model predictions, four case studies are considered with various magnetic particles and matrix materials. Actuation performance is measured experimentally to validate the model for the case studies. The results show good agreement between experimental measurements and model predictions. A further parametric study is conducted to investigate the effects of the magnetic properties of particles and external magnetic fields on the free deflection. In addition, complex shape programming of the unimorph actuator is demonstrated by locally altering the geometric and material properties.

Magnetoactive elastomers (MAEs) are capable of large deformation, shape programming, and moderately large actuation forces when driven by an external magnetic field. These capabilities enable applications such as soft grippers, biomedical devices, and actuators. To facilitate complex shape deformation and enhanced range of motion, a unimorph can be designed with varying geometries, behave spatially varying multi-material properties, and be actuated with a non-uniform external magnetic field. To predict actuation performance under these complex conditions, an analytical model of a segmented MAE unimorph is developed based on beam theory with large deformation. The effect of the spatially-varying magnetic field is approximated using a segment-wise effective torque. The model accommodates spatially varying concentrations of magnetic particles and differentiates between the actuation mechanisms of hard and soft magnetic particles by accommodating different assumptions concerning the magnitude and direction of induced magnetization under a magnetic field. To validate the accuracy of the model predictions, four case studies are considered with various magnetic particles and matrix materials. Actuation performance is measured experimentally to validate the model for the case studies. The results show good agreement between experimental measurements and model predictions. A further parametric study is conducted to investigate the effects of the magnetic properties of particles and external magnetic fields on the free deflection. In addition, complex shape programming of the unimorph actuator is demonstrated by locally altering the geometric and material properties.
Keywords: magnetoactive elastomer, unimorph actuator, segmented analytical model (Some figures may appear in colour only in the online journal) * Author to whom any correspondence should be addressed.
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Introduction
Conventional medical devices such as ventilation masks and foot orthotics are typically passive and static, which means they cannot accommodate patient-specific requirements or shape changes as treatment progresses. If a shape change is needed to accommodate patients' growth or healing or to minimize pressure sores, the device must be replaced in consultation with a physician. To efficiently resolve these issues, smart materials are an attractive option for their shape-programming capability under external stimuli. The shape change responses to external stimuli make smart materials promising candidates for biomedical and other devices such as self-expandable stents [1], soft grippers [2,3], battery actuators [4], bistable compliant mechanisms [5], bionic locomotion [6][7][8], isolator [9] and acoustic metamaterials [10].
Fabrication of tunable adaptive devices using smart materials with complex configurations is highly desirable to realize as-designed features and satisfy user requirements. To meet these needs, 4D printing has been developed as a type of additive manufacturing (AM) in which the fabricated structures can be customized and change shape over time, in contrast with conventional manufacturing methods that fabricate relatively simple geometries because of limitations such as the complexity of manufacturing elaborate molds. The end products take advantage of layer-by-layer manufacturing to realize intricate configurations, tuning of local material properties, and diverse shape programming capabilities. For example, Cao et al [11] used AM to fabricate a magnetic soft material consisting of thermoplastic rubber and carbonyl iron particles (CIPs) with complex geometries and programmed actuation that mimic octopus tentacles, butterflies, and flowers. Ma et al [12] focused specifically on property tuning, demonstrating that the magnetization of the pre-magnetized magnetic shape memory polymers (SMPs) was tunable by reorientating during post-magnetization.
Another attractive benefit of applying AM to fabricate smart materials is the capability of fabricating functionally graded materials by combining multiple materials with different compositions. One study [13] applied a fused deposition modeling process to fabricate the functionally graded SMP to perform self-folding, coiling, and deploying. Qi et al [14] fabricated functionally graded magnetorheological polymers consisting of polycaprolactone (PCL) and thermoplastic polyurethane to respond to multiple external fields simultaneously. Previous work from the authors demonstrated the fabrication of functionally graded magnetic SMPs using a reactive extrusion AM process that mixes epoxy resins, hardeners, and magnetic particle concentrations on demand [15].
Magnetoactive elastomers (MAEs) consisting of soft elastomer matrices and magnetic particles are unique because of their responses to contactless magnetic fields. The choice of elastic matrices incorporates functionalities of MAE to respond to temperature, UV exposure, pH, electric field, etc [16,17]. Depending on the application, hard and soft magnetic particles can be employed with varying magnetic properties to integrate fast, reversible, and reprogrammable actuation [18]. Hard magnetic particles (HMPs) feature high coercivity and high remanent magnetization and can be poled during curing to form a preferred magnetization orientation. Upon application of an external magnetic field, this predetermined orientation attempts to align with it to minimize the Zeeman energy [19]; these magnetic particles with programmed magnetization orientation are subjected to micro-torques that stimulate macro-deformations of the MAE when under an external magnetic field. In contrast, soft magnetic particles possess low coercivity and little to no remanent magnetization. However, they can also be actuated to realize elongation, contraction, and coiling modes by constraining the boundaries of the samples and designing the applied magnetic field at different stages and have been demonstrated as artificial muscle actuators for their favorable stress-strain capabilities, flexibilities, and actuation frequencies [17,20]. In addition, soft magnetic particles can be applied on remote heating under a high frequency alternating current magnetic field by Eddy current heating and by hysteresis loop, which has the potential to heat a thermosensitive polymer above its glass transition temperature and soften it [17,18]. Subsequently cooling and locking the shape manipulation into place maintains functional performance after the external stimuli are removed and offers the added benefit of avoiding continuous energy consumption.
To reduce the need for trial-and-error experimentation, models can aid the shape programming process by predicting the actuation performance of compliant structures made of stimuli-responsive smart materials. In general, researchers can select theoretical models (e.g., analytical and kinematics models), or finite element models based on the required fidelity, design stage, and practical conditions [19]. Finite element simulation is most widely used as a high-fidelity model. It can be implemented effectively for flexible materials with complex geometries, but the computational cost can be relatively high [21][22][23]. On the other hand, lower-fidelity theoretical models are limited to relatively simple geometries, but they are capable of predicting the stress and deformation of active materials with low computational costs. For example, for a laminated cantilever structure, the difference in free deflection between a 1D analytical model and a 3D FEA was 4%, which lowered the necessity to apply a complete 3D FEA [24]. Energy minimization methods have been studied widely for MAEs by considering the magnetostatic, stress-induced, and magnetocrystalline energies [25,26]. The energy methods have been shown to be valid for 1D laminated structures through integration with traditional beam theory [24] and for 2D structures by coupling classical plate theory with Poisson's effect [27]. A magneto-induced actuation model was also established for an isotropic MAE by taking account of hyperelastic properties, and the results revealed the magneto-induced actuation stress and shear modulus increased with the content of magnetic particles (e.g., CIPs) [28]. A segment-wise modeling approach makes it easier to account for local anisotropy in material properties or non-uniformness of geometry. For example, a 1D constitutive model was proposed for a selffolding bimorph consisting of both MAE and electroactive terpolymer, which was actuated under multiple fields [29]. The segment-wise approach has the potential to predict the behavior of smart materials fabricated with AM, which enables localized changes in material properties and shapes but also often leads to process-induced uncertainties in the asmanufactured part.
With the rapid development of AM and its application in 4D printing, multi-material adaptive structures can be fabricated with complicated geometries [13], switchable properties [12], and versatile functionalities [14]. It is essential to develop the capability to model and optimize the performance of structures by accounting for the characteristics of AM-printed samples and associated multi-field actuation to meet different shape change requirements. Previous work has focused more on the structural design of conventionally fabricated composites [30] or AM-printed samples with single materials [31], which either lack geometric complexity or respond to only one actuation mechanism. Limited modeling and design work is focusing on AM-fabricated, functionally graded materials that consider multi-field actuation and anisotropic local properties.
The goal of this research is to model an MAE unimorph consisting of conventionally fabricated or AM-printed materials. Four case studies are presented that illustrate the model for hard and soft magnetic particles and different matrix materials. The model is validated using experimental measurements of the actuation performance of these multi-material unimorphs in response to external thermal and magnetic fields. Free deflection, representing the maximum displacement of a MAE unimorph under specific external stimuli is selected to validate the model.

General model approach
A unimorph actuator consists of a layer of active material bonded to a passive substrate. The unimorph is modeled using beam theory, assuming large deflections and linear elastic material behavior, where it is divided into N segments of equal length to account for nonuniform geometry and anisotropic material properties along the length. This segmented beam approach was initially proposed by [32] for electroactive polymer actuators and further developed by [4] for electrochemical actuation. Figure 1 shows a schematic to illustrate the beamlike unimorph consisting of an active MAE and a passive substrate that are discretized equally into N segments. Within each segment, the width and thickness are assumed to be uniform for both materials. An ideal bonding condition between two layers is assumed, i.e., there is no relative displacement at the interface between layers under the external field.
Under the actuation of an external magnetic field, the MAE rotates to align with the direction of the magnetic field. However, because of the passive layer that constrains the deformation, a subsequent bending actuation is developed with a potential axial extension from the strain mismatch between active and passive layers. Therefore, the strain in segment i, ε i , comprises extensional strain along the interface ε exi and bending strain K i x, where K i is the induced free curvature on account of the actuation as shown in equation (1), Force and moment equilibrium equations are considered to determine the two strain components for each segment. The governing equations are shown in equations (2a) and (2b), where N layer denotes the total number of layers. Subscripts i and j denote the segment number and layer number, respectively. Two layers are used in this study to represent the active and passive layers of a unimorph. To determine the force and moment, the stresses are integrated over the layer thicknesses, where the location of the neutral axis is defined by x 0 (figure 1). The resultant force and moment equilibrium equations are expressed as equations (3a) and (3b), where E a and E p are the elastic moduli, and t a and t p are the layer thicknesses of the active and passive layers, respectively. The effect of the magnetic field is represented by an effective torque acting on segment i, τ i which is described in section 2.2.
The relationship between the local deflection angle θ locali and free curvature K i of each segment is derived based on the Bernoulli-Euler equation, as shown in equation (4), where θ locali is defined in the local coordinate system of each segment and s i is the segment arc length. The curvature is known from solving equations (3a) and (3b). By separation of variables, θ locali is determined as equation (5), where arc length s i = l i is used to account for large deformation. The segment's global deflection angle θ i is expressed in terms of the local deflection angle as in equation (6), The determined free curvature and deflection angle are used to determine the local deflection of each segment. By applying the chain rule of differentiation to equation (4), the local deflection components of each segment, denoted as δ zlocali and δ xlocali , respectively, are expressed in equations (7a) and (7b), Coordinate transformation is used to convert the local deflections δ zlocali and δ xlocali , to global quantities, as shown in equation (8a). The displacements between two ends in each segment are defined as X li and Z li in a global coordinate system. The resultant global deflection components, denoted as X i and Z i (as shown in figure 2), are expressed in equations (8b) and (8c),

Soft and HMPs
The effect of the applied magnetic field is modeled using an effective torque on each segment τ i , which depends on the magnetic moment of the material and the applied magnetic field, both of which are vector quantities, as in equation (9), where µ 0 is the permeability constant, H i is the external magnetic field applied to segment i, and m ci is the magnetic moment of the MAE unimorph at segment i. For the segmented approach implemented in this study, the effective torques are assumed to be acting at each segment. The magnitude of the effective torque τ i on segment i is shown in equation (10), where H i depends on the location of the segment. φ i is the angle between the direction of m ci and H i , which is a nonconstant value in each segment and needs to be determined segment by segment. θ i is the complement angle of φ i , as shown in figure 2. m ci is determined in equation (11), Where M ci is the magnetization (in emu/cm 3 ) of the MAE, V ci is the effective volume, and f i is the volume fraction of magnetic particles. m mHi represents the mass magnetization (in emu/g) of the magnetic particles at segment i. It is assumed m mHi is the remanent mass magnetization for HMPs independent of the external magnetic field since the difference in remanent magnetization and saturation magnetization is small for HMPs. Published work has demonstrated that this assumption is valid for Barium hexaferrite (BHF) particles [22,33,34]. ρ Hi is the density of the magnetic particles of segment i. The induced dipole-dipole interaction is assumed to be small enough in comparison to the dominant magnetic response. For an MAE with HMPs like BHF (see section 3.3.4), the material is poled with a strong magnetic field during curing to form a preferred magnetization (M c ) orientation. The preferred orientation aligns with the direction of the external magnetic field, which generates a torque (τ ) acting on the unimorph.
For an MAE with soft magnetic particles, the actuation mechanism is fundamentally different. Soft magnetic particles do not have a preferred magnetization orientation and the remanent magnetization of soft particles is much lower than that of hard particles [18]. To estimate the magnetic moment, it is assumed that the mass magnetization is a function of the external magnetic field (H) according to the hysteresis behavior, as shown in equation (12), The dependence of m m Hi on H is determined experimentally, as described in the case studies. For either soft or HMPs, the initial θ 1 is set to zero to determine the effective torque τ 1 using equation (10), and further determine the K 1 and θ 1 with the governing equations (3a), (3b), (5) and (6). Then the new θ 1 is updated and used to determine effective τ 2 and K 2 . Those steps are iterated until the last segment is reached.

Materials and manufacturing process
Two methods, casting and AM, are used to fabricate MAE unimorph, as described in sections 3.1.1 and 3.1.2, respectively.

Casting method.
To fabricate the MAEs for case studies A, B, and D, a mechanical stirrer and bath sonicator are used to mix the particles (BHF or iron oxide) in the amount of 30 wt% with respect to the polymer matrix solution. The mechanical stirrer is an IKA RW 20 digital with a maximum spinning rate of 2000 rpm, and the bath sonicator is manufactured by Fisher Scientific with 2.8 l volume. The combination of mechanical stirrer and bath sonicator ensures that the solution is uniformly mixed, yielding good dispersion and no agglomeration of particles. After the mixing, the solution is degassed again in a vacuum to ensure all air bubbles are eliminated. For PDMS-based samples, the curing agent is then added after degassing at a weight ratio of 1:10 curing agent to base. The solutions are cast using a doctor blade or poured The fabrication processes of BHF-PDMS and iron oxide-PDMS composites vary due to their different magnetic properties, as shown in figure 3. For HMPs (such as BHF), two sets of permanent magnets are used to polarize the particles as the PDMS cures. The fully cured MAEs are then cut into geometric shapes, such as rectangles and squares, for magnetic actuation in order to systematically study the behavior and performances of the materials.
Similar fabrication processes are used to prepare iron-oxide PVA samples. PVA pellets (Alfa Aesar) are dissolved in distilled water in 12 wt% using a hot plate (Corning, PC-420D) and magnetic stirrer. The beaker containing the PVA solution is submerged in a silicone oil bath (Clearco Products Co., Inc.) which is maintained at 90 • C using a temperature probe. The magnetic stirrer is set at 400 rpm and the PVA pellets are slowly added into the solution over the entire course of mixing to avoid large clustering. After an hour of continuously mixing the solution, the beaker is sealed using plastic wrap (Bemis™ Parafilm™ M) and aluminum foil (Reynolds) and is allowed to sit overnight to allow any air bubbles to escape from the solution. The flowchart in figure 4 summarizes this process. It is important to note that weight percentage (for example, 30 wt% of iron oxide) is calculated based on the weight of the composite, where PVA is 70 wt%. The iron oxide-PVA thin film exhibits a thickness of approximately 0.05-0.1 mm which is significantly thinner than iron oxide-PDMS (approximately 0.3-0.5 mm).

Reactive extrusion additive manufacturing (REAM)
method. REAM is used to create a laminated MAE unimorph consisting of a layer of iron oxide filled PDMS and a layer of passive epoxy. REAM systems function by metering liquid thermoset resins and catalyzing agents at the appropriate ratio through separate transport lines before combining the two in a passive mixing nozzle. The passive mixing nozzle then mixes the resin and catalyst together to the point of homogeneity immediately prior to deposition, after which the mixture is allowed to cure in situ [35]. The REAM system used in this work consists of a Lulzbot Workhorse desktop gantry FDM machine retrofitted with a REAM material delivery system. The material delivery system in turn consists of a carriage to which resin and catalyst reservoirs, peristaltic metering pumps, and a static mixing nozzle are mounted [15].
Prior to use in the REAM system, the epoxy (EPON 8111 and EPIKURE 3271) is thickened by mixing 3.5 wt% fumed silica into both the resin and catalyst, after which both are allowed to degas naturally. This is necessary in order for the freshly deposited epoxy to retain its shape prior to curing. The iron oxide filled PDMS is prepared for use in the REAM system using the same method outlined in section 3.1.1 with one modification: 3.5 wt% fumed silica is added along with the iron oxide, producing a more thixotropic feedstock capable of retaining as-deposited geometry. The laminated unimorph is created using a simple alternating raster deposition pattern with 1 mm tall, 1.8 mm wide rasters, and a 2 mm diameter nozzle to ensure raster overlap. After heating the build plate to 90 • C, the epoxy layer is deposited first as its gel time is roughly an order of magnitude shorter than that of the PDMS, allowing for more rapid fabrication. After deposition of the epoxy layer, the feedstock reservoirs are swapped out for fabrication of the iron oxide filled PDMS. The PDMS is then deposited atop the epoxy and allowed to sit on the heated build plate for 10 min to ensure the gelation of the PDMS.

Material characterization
The mechanical properties of the MAEs are characterized through tensile tests. Samples for the tensile tests are cut into 1 × 4 cm specimens, and 4-5 samples are prepared for each batch of samples for accuracy. The samples are elongated at a rate of 5 mm min −1 with a maximum force of 48 N. The amount of force applied to the sample is recorded every 0.1 s until failure. Engineering stress values are calculated with force divided by cross-sectional area, and strain values are calculated using the crosshead length divided by the width of the sample. A stress-strain curve is then obtained to calculate Young's modulus using the slope of the elastic portion of the curve. Miniature tensile testing is also conducted on different samples to acquire Young's modulus with a strain rate of 0.1 mm s −1 at room temperature.
To determine the magnetic properties, vibrating sample magnetometry (VSM) is performed on both magnetic microparticles and magneto-active polymer-based unimorph. The samples are placed in a uniform magnetic field which induces a magnetic moment in the samples. A sinusoidal motion in the field is applied to the sample, and sensing coils around the sample then measure the magnetic flux density, such that the relationship between the magnetization of the sample and the applied field can be quantified. A hysteresis loop is then plotted to represent the data. The saturation magnetization, which is the highest magnetization detected from the sample, is then compared among samples to analyze the magnetic properties. Along with saturation magnetization, coercivity, and remanent magnetization are observed to characterize both hard and soft magnetic properties. The hysteresis loop of the MAE can also be considered by considering the volume fraction/weight percentage of iron oxide particles. The VSM-based methodology to characterize both the particles, the composites, and the determination of volume content based on VSM, is described in our prior work [15], Appendix section.

Model applications for four case studies
Here, four case studies are introduced to illustrate the impact of MAE unimorph with different magnetic particles or mechanical properties, as well as the impact of the magnetic field, on the actuation performance of the unimorph. Results are reported in section 4. Table 1 summarizes the geometry, material properties, and external field conditions, which are inputs to the model in the different case studies. The width (w) is 1 cm for all four case studies. In particular, constant Young's modulus of the active MAE are assumed for all four cases without considering the influence of magnetic field on the elastic properties of MAE [36][37][38][39].

Case study A: PDMS with iron oxide and a permanent
magnet. In this case study, iron oxide particles, which are soft magnetic particles, are uniformly dispersed within PDMS, and the film is then cured. Scotch tape (Magic Tape, 3M) is applied as the passive substrate. To account for the dependence of the mass magnetization of soft magnetic particles on the magnetic field, the hysteresis loop of pure iron oxide  particles is measured via VSM as shown in figure 5. From this data, two functions are fit to describe the dependence of m mH on H, as noted in figure 5 [40]. Although the MAE unimorph is magnetically anisotropic due to the demagnetizing field and the dipole-dipole interaction [41], the direction of the induced magnetization for the beam-like MAE unimorph is assumed to be aligned with the length of the unimorph from tip to base. For this case study, a NdFeB Grade 52 cylindrical permanent magnet (K&J Magnetics, Inc.) with a length of 5 cm and a diameter of 5 cm is used to provide the external magnetic field. This field is spatially nonuniform, and the magnitudes are measured experimentally using a Gaussmeter (PCE-MFM 3000, PCE Instruments) at spatial intervals of 0.5 cm in both axial and transverse directions around the magnet. The direction of the field at each spatial location is obtained using publicly available data from the manufacturer [42]. The direction of the field is superimposed on the magnitude data at an interval of 0.2 cm along both directions. The external field H as a function of the location in the unimorph is considered for each segment. The closest magnetic field data point (direction and magnitude) to one unimorph segment after deflection is selected to represent the external field H applied to that segment.

Case study B: PVA with iron oxide and a permanent
magnet. This case study considers iron oxide with PVA as a matrix, which is similar to the previous case (iron oxide with PDMS) in terms of modeling m mH and H, where the difference is the elastic modulus of the PVA sample. PVA is much stiffer than PDMS (on the order of 10 3 times stiffer as shown in table 1) so a different actuation performance is expected with less deflection.

Case study C: AM fabricated epoxy and PDMS with iron oxide and a permanent magnet.
Case study C considers an AM fabricated unimorph, with MAE consisting of uniformly dispersed iron oxide particles embedded in PDMS, while the passive substrate is epoxy, which also exhibits shape memory behavior. The AM process is described in section 3.1.2. The magnetization m mH of the iron oxide particles and the magnetic field H of the permanent magnet is modeled in the same way as in case studies A and B. However, due to the nature of AM, the variation in geometry (e.g. thickness) of the unimorph and the heterogeneity of the material properties must be taken into account. It is observed that the active MAE and passive epoxy layer have variable thicknesses, resulting in a wavy interface between the two materials as shown in figure 6. To account for this variability in the model, the thickness of the AM-fabricated MAE unimorph is measured at different locations along the length of the unimorph for both active and passive layers with a tabletop micrometer. The mean value of multiple measurements is applied to each segment in the model ( figure 6). To soften the shape memory epoxy, the MAE unimorph is heated in-situ with a heat gun to maintain temperatures around 110 • C during actuation process. Hence, the Young's modulus of epoxy above glass transition temperature is considered in table 1.

Case study D: PDMS with BHF and an electromagnet.
Case study D focuses on HMPs, BHF. A uniform dispersion of BHF particles inside a PDMS matrix is assumed. For pure BHF particles, the magnitude of the mass magnetization, m mH , is assumed to be constant at 40 emu g −1 , which is the average value of the measured remanent magnetization for BHF particles [43,44]. The direction of m mH is based on the poling direction, which is perpendicular to the length of the unimorph and consistent with the cast samples in the experiment.
In this case study, a C-shaped electromagnet (ranging from 7 to 260 mT) is used to actuate the sample where the magnitude of field strength, H, is controlled by a DC power supply (72-6180 A, TENMA). A voltage ranging from 0 to approximately 15 V can be input where larger voltages cause larger magnetic fields. The direction of H is assumed to be uniform, directed from the top pole face to the bottom pole face. The magnitude of H (z) at different spatial locations is measured using the Gaussmeter, and the measured data at different voltages are fitted with quadratic functions [45].

Actuation performance quantification
The unimorph samples are actuated under either the permanent magnet or C-shaped electromagnet with one fixed end. For Case C, because of the characteristics of the shape memory epoxy, the samples are heated in-situ with a heat gun to maintain temperatures around 110 • C during actuation to achieve the soft behavior. The deflection is recorded and actuation is quantified with ImageJ. For all of the case studies, the degree of bending is quantified as the free deflection (C) with respect to the original position as shown in figure 7(a). At least three measurements are recorded for each sample. For cases using the permanent magnet, the distance d denotes the distance between the original position of the free end of the unimorph and the pole surface of the permanent magnet ( figure 7(b)); hence, a shorter distance indicates larger magnetic field strength and vice versa.

Case study A.
The experimental actuation performance of case study A is shown in figure 8(a). It can be seen that the original unimorph sample is not straight, prior to applying the magnetic field, due to the existing prestress during the manufacturing process. For this reason, the displacement in response to magnetic field to the initial stage is measured and compared to eliminate the effect of initial sample deformation. In the model, a straight sample is assumed initially. As the distance, d, between the magnet and sample increases from 3 cm to 6 cm, the deflection, C, of the bottom tip of the unimorph decreases because a lower magnetic field weakens the effective torque acting on the unimorph. Figure 8(b) illustrates the model-based predictions of the free deflection at different distances. When visually comparing figures 8(a) and (b), a good qualitative match between experimental results and model-based predictions is observed, and a quantitative comparison is shown in figure 8(c) and table 2. It can be seen that the model prediction is within the standard deviation of the experiment results, which demonstrates the accuracy of the models for this case study.

Case study B.
Case study B uses PVA as the matrix material, which has a higher modulus than PDMS (table 1). Similar to case study A, the magnitude of the magnetic field strength is controlled by the distance, d, between the samples and the magnet. Larger free deflections, C, are detected with decreasing distance in both experimental results (figure 9(a)) and model predictions ( figure 9(b)). Differences between measurements and model predictions, especially at larger external field conditions, are partially attributed to the initial curvature of the unimorph ( figure 9(a)). Since the model assumes an initially straight sample ( figure 9(b)), the location of the modeled MAE unimorph is different in comparison to the measured unimorph as well as the corresponding magnetic field in terms of direction and magnitude. The free deflection (figure 9(c) and table 2) shows a good agreement between experiment measurement and model predictions, which validates the model for case study B.
MAE with PVA, with a higher elastic modulus (table 1), is expected to exhibit less free deflection compared to MAE with PDMS. However, due to the solvent-based casting, the layer thickness of the PVA sample is about half of that of the PDMS in case study A, and it is difficult to realize consistent thickness between samples. A normalized deflection, defined as the model-predicted free deflection divided by the active layer thickness (table 1), is therefore determined and listed in table 2. It confirms that a less normalized deflection is performed with MAE with PVA compared to MAE with PDMS under the same external field condition.

Case study C.
For AM unimorphs consisting of active PDMS embedded with iron oxide particles and passive epoxy, experimental measurements and model-based predictions of free deflection are shown in figures 10(a) and (b), respectively, under different magnetic field strengths represented by the distance between the sample and the permanent magnet. Similarly to previous samples, the AM laminated unimorph exhibits higher free deflection with higher magnetic field strength, which is expected due to a larger effective torque acting on the sample. Figure 10(c) and table 2 quantitatively compare the model-based predictions of the free deflection with the experimental measurements of three samples. Results show that good agreement is obtained between experimental results and model-based predictions at lower magnetic fields, whereas at higher field strength (d = 1, 1.5, 2, 2.5 cm) the model-based predictions slightly overestimate the free deflections. Instead of considering the variation of thickness along the width at each segment (i.e., the wavy interface between active and passive layers), the average thickness is applied to the model, which could contribute to the error since layer thickness both affects bending stiffness and effective torque acting on the unimorph.

Case study D.
Case study D considers the unimorph of an MAE with HMPs, BHF, under the electromagnet. Figures 11(a) and (b) illustrates the experimental and modelbased results of free deflection of the MAE unimorph, respectively, at different voltages. Higher free deflection is expected with higher voltages since larger magnetic fields generate stronger magnetic torques on the unimorph, which is consistent with the model results. Figure 11(c) and table 2 show the model validation results for the displacement, C, of the bottom tip. The comparisons between the measured and modeled results show good agreement at lower voltages. However, discrepancies are observed at higher voltages where the model predictions overestimate the measurements. An inconsistent layer thickness along the unimorph, as well as existing small pores during manufacturing, could contribute to the differences, which also affect the effective torque acting on the unimorph. The effect of the inconsistent layer thickness on the effective torque is magnified with higher voltages since the H i is higher according to equation (10). Another possible reason could be the variation of field strength along the X direction ( figure 11(a)). When considering the spatial distribution of the magnetic field strength, only the variation along the Z direction is accounted for. To investigate this hypothesis, more gauge readings of field strength near the edge of the pole face are performed, and the lower field strength is measured in comparison to the central region, which could explain the relative lower tip displacement measurements since the samples move closer to the edge of the pole face when deforming.

Effect of magnetization and external magnetic field
The proposed segmented beam model has been validated for four case studies, and in this section, it is further implemented to conduct parametric studies to investigate the effect of distinct parameters. The parametric study is helpful to support our future design and optimization work and to guide the experimental work to reduce trial-and-error. Here, the effect of magnetization in terms of magnitude and direction and the effect of the external field strength are investigated. The parameters used for this analysis are listed in table 3.
For the magnitude of magnetization as listed in table 3, 40 emu g −1 (value measured for BHF particles) is used for the constant mass magnetization (m mH ) case, while the fitting functions for iron oxide particles in figure 5 and table 3 are used to estimate the magnitude depending on the magnetic field strength for the varying mass magnetization case (m mH (H)). Two directions of magnetization are assumed where one is perpendicular to (⊥) the unimorph ( figure 12(a)) and the other is parallel to (//) the unimorph ( figure 12(b)). Figure 12(c) illustrates the effect of magnetization in terms of magnitude and direction on the free deflection. A uniform external field is assumed when exploring the effect of magnetization, where the magnitude is assumed to be 0.15 T and the direction is unidirectional from left to right (figures 12(a) and (b)). First, in terms of the varying direction of mass magnetization with a constant magnitude, 40 emu g −1 , a larger effective torque is realized for magnetization parallel to the beam since the angle between the magnetization and the magnetic field is closer to 90 • . Therefore, larger free deflection is observed as the direction of magnetization attempts to align with the magnetic field direction (figure 12(c) blue solid or green dash curve). However, for the case in which the magnetization is perpendicular to the beam, it is more difficult for the unimorph to align with the direction of the magnetic field; hence, the magnitude of the deflection is smaller, and the beam     figure 12(c)) due to the difference in effective torque.
In the cases where the magnetic field is non-uniform, three conditions are considered: only the magnitude of the field strength (H mag (X, Z)) is varied with a uniform direction (from left to right), only the direction of the field strength is varying (using the spatial distribution of direction reading from the manufacturer website [42]) (H dir (X, Z)) with a constant magnitude (0.15 T), and both the magnitude and direction are varied (H mag,dir (X, Z)) at different locations. The spatial distribution of the magnetic field due to the permanent magnet used for the experiment is applied to these varying field conditions where magnitude is measured with Gaussmeter and direction is acquired from the manufacturer's website [42]. Both constant and varying magnitudes of magnetizations are considered while the direction is assumed to be parallel to the beam. The model results for varying field cases are illustrated in figure 12(d), which shows that the predictions of free deflection depend on both the magnitude and direction of the field since both impact the magnetization and therefore the effective torque acting on the segments. With the permanent magnet as the actuation field where both magnitude and direction vary, the case considering both direction and magnitude difference (purple triangle in figure 12(d)) exhibits the most accurate prediction result. When only the magnitude of field strength is considered, the direction of magnetization ( figure 12(b)) tends to align with the external field (from right to left), and free deflections perform as curves (blue solid curve and red asterisk in figure 12(d)). In addition, the magnetization increases with the locations with larger field strengths when the unimorph is bent towards the magnet. The free deflection is straighter when only the direction of the external field is taking account since the direction of magnetization is easier to align with the direction of H. The importance of the sensitivity study is to comprehend the difference in predicted free deflection with a varying magnetic field. Understanding the precise magnitude and direction of the magnetic field at different spatial locations is significant to predicting the behavior of these unimorphs.

Shape programming by varying the shape and material properties of the MAE unimorph
The analytical model is useful to predict the actuation performance of the MAE unimorph and can be extended to program application-specific shape changes. In this section, shape programming of the unimorph is considered by varying the geometries or material properties of the MAE material. To be specific, the nonuniform segment-wise active layer thicknesses of the active material (MAE), volume fractions of active particles, and magnetization are selected to achieve complex actuated shapes. These heterogeneous structures are possible to realize via our REAM process in the future. PDMS with BHF is selected here as active material and it is actuated under electromagnet when the voltage is 14.7 V. Single variable approach is applied here where the default active layer thickness is 0.3 mm, the default volume fraction of BHF particle is 0.1 and the default magnitude of mass magnetization is 40 emu g −1 . For other parameters refer to table 1 for Case D. Figure 13 shows the actuated shape with a varying active layer thickness of the MAE material. It can be seen that the free deflection in figure 13(a) increases with a thinner layer thickness in most of the selected layer thicknesses. However, the free deflection at t = 0.3 mm is larger than that at t = 0.1 mm, which emphasizes the tradeoff between bending stiffness and the generated effective torque with varying active layer thickness. That is, increasing the layer thickness increases the bending stiffness of unimorph, yet it also increases the amount of magnetic particles assuming the volume fraction of particles remains the same. The effect of bending stiffness is stronger in most of the selected thicknesses than the influence of effective torque. Figure 13(b) illustrates the unimorph with nonuniform active layer thickness from base to tip. A thinner layer thickness (0.1 mm) near the base results in a large local curvature (blue solid and yellow dot curve in figure 13(b)), hence the whole unimorph becomes flat where the direction of magnetization aligns with the external field. The thicker layer thickness near the base reduces the local curvature, but the remaining unimorph can still perform a flat shape with a thinner thickness near the middle section of the unimorph (red asterisk in figure 13(b)). The gripper configuration can additionally be designed with a careful selection of layer thickness, as demonstrated with the green dashed and purple dot curves in figure 13(b).
Magnetization plays an important role in shape programming, especially for MAE with HMPs which have a preferred orientation aligning with the magnetic field.
A larger magnitude of mass magnetization always induces higher effective torque acting on the MAE unimorph, which further contributes to a larger free deflection ( figure 14(a)). The direction of mass magnetization affects the bending direction relative to the direction of the external magnetic field. As shown in figure 14(b), if only the magnitude of the mass magnetization is variable and is altered at different locations, the MAE unimorph bends towards the same direction (yellow dot and red asterisk). By alternating the mass magnetization direction along the length of the unimorph, a 'S' like shape is obtained (blue solid and green dashed curve). This case demonstrates the possibility of complex shape programming via anisotropic magnetization, which can be further refined using design optimization of the structure.

Conclusions
In this work, a segmented beam model is developed to predict the actuation performance of an MAE unimorph actuator considering large deformation. The model accounts for MAEs consisting of active magnetic particles with passive substrates actuated under a magnetic field. The effective torques acting on each segment are created by aligning the magnetization direction to the external field direction, as governed by the segmented beam model. The main differences in the model between hard and soft magnetic particles are the assumptions made for magnetization determination. For HMPs like BHF, the magnetization has a constant magnitude and a predetermined direction established by the poling process, which is equivalent to the remanent magnetizations. For soft magnetic particles, such as iron oxide, the dependence of the magnitude of the induced magnetization on the external field is taken into account, and the direction of the induced magnetization under the magnetic field is assumed to be aligned with the beam, which changes in response to the magnetic field. The model is validated for case studies considering unimorph actuators with soft or HMPs, which are fabricated by both conventional casting and AM. The model is shown to have a good agreement of free deflection with experimental measurements for the four case studies. The validated analytical model is further applied to parametric studies to investigate the effects of magnetization and the external field on free deflection. Results reveal that both the magnitude and direction of the magnetization affect the free deflection under an external field substantially. In addition, the model-based predictions confirm that knowledge of the spatial distribution of the magnetic field is necessary to obtain accurate predictions of the deflections of these materials. Also, by locally altering the layer thickness of the MAE or the volume fraction or magnetization of the active particles, complex shapes of unimorph can be programmed. Nonuniform curvatures are demonstrated, which could be useful for designing and optimizing structures for patientspecific applications as part of future work. A cyclic use of the envision application will also be consider as the future work [46][47][48].

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).