A concept recently proposed by the authors is that of a multifield sheet that folds into several distinct shapes based on the applied field, be it magnetic, electric, or thermal. In this paper, the design, fabrication, and modeling of a multifield bifold are presented, which utilize magneto-active elastomer (MAE) to fold along one axis and an electro-active polymer, P(VDF-TrFE-CTFE) terpolymer, to fold along the other axis. In prior work, a dynamic model of self-folding origami was developed, which approximated origami creases as revolute joints with torsional spring–dampers and simulated the effect of magneto-active materials on origami-inspired designs. In this work, the crease stiffness and MAE models are discussed in further detail, and the dynamic model is extended to include the effect of electro-active polymers (EAP). The accuracy of this approximation is validated using experimental data from a terpolymer-actuated origami design. After adjusting crease stiffness within the dynamic model, it shows good correlation with experimental data, indicating that the developed EAP approximation is accurate. With the capabilities of the dynamic model improved by the EAP approximation method, the multifield bifold can be fully modeled. The developed model is compared to the experimental data obtained from a fabricated multifield bifold and is found to accurately predict the experimental fold angles. This validation of the crease stiffness, MAE, and EAP models allows for more complicated multifield applications to be designed with confidence in their simulated performance.

## Introduction

Self-folding origami has been the subject of considerable research recently, resulting in many novel applications and actuation approaches [13]. In most cases, a single active material is used to actuate a model with bending or folding that can be permanent [4,5] or reversible [69]. Each active material used has advantages and disadvantages over the others, and properties of many of these materials have been evaluated and compared [10,11]. While more difficult to design, fabricate, and model, multifield origami combines multiple active materials into a single design, enhancing the sophistication of actuation and control [12,13]. A concept recently proposed by the authors is that of a multifield sheet that could fold into several distinct shapes based on the applied field, be it magnetic, electric, or thermal [12]. This sheet could also unfold back to the flat state once the applied field(s) are released.

In this paper, the design, fabrication, and modeling of a multifield sheet known as the bifold are presented, pictured in Fig. 1. The bifold utilizes P(VDF-TrFE-CTFE) terpolymer [14,15], an active material that responds to applied electric fields, and MAE [16,17], an active material that responds to applied magnetic fields, to fold along two distinct crease lines. This multifield sheet was designed such that the fabrication and experimental characterization would be straightforward, minimizing the kinematic complexity to focus on the active material models themselves. Such a design allows for ready comparison to a model of the bifold as a first step to achieve the simulation, design, and fabrication of more complicated multifield sheets.

Fig. 1
Fig. 1
Close modal

There are several different types of origami models that can be used to simulate folding and self-folding: kinematic, analytical, dynamic, and finite element. To simulate the multifield bifold, the selected model type must be capable of simulating the effects of both magnetic and electric fields, in addition to the bending stiffness and damping inherent in origami creases. In this work, a “crease” is generally considered to be a compliant joint with a radius of curvature. In prior work, a method for the dynamic modeling of self-folding was detailed that successfully modeled the effects of crease stiffness, crease damping, and magneto-active elastomer on the folding of origami-inspired designs [10,18]. These models solved the self-folding problem very quickly by approximating origami panels as rigid links, origami creases as revolute joints and torsional spring–dampers, and magneto-active elastomer as torques applied to the panels. While direct numerical simulation as would be possible in multiphysics-capable finite element software may provide a more accurate approximation, the faster solution time of a dynamic model is beneficial during the development of self-folding designs.

While a suitable approximation of MAE for use in dynamic models has been demonstrated, an approximation for terpolymer is not yet available. In this work, a method for approximating the effect of an EAP actuator (such as terpolymer) on an origami-inspired design is proposed and validated using experimental data from a terpolymer-actuated barking dog origami model [7]. Methods for approximating the behavior of origami creases in the dynamic model are also presented. The design and fabrication of the multifield bifold are then discussed, and an experiment is detailed by which twofold angles from the fabricated samples are measured. Using the developed and validated EAP approximation, the dynamic model of the bifold is compared to experimental data. With a successfully validated multifield model, more complicated multifield sheet designs can be simulated with confidence in their accuracy.

## Background

### Multimaterial Self-Folding Designs.

While there are many origami-inspired designs that utilize active materials for actuation, there are relatively few that utilize multiple active materials in the same design to accomplish multifield actuation.

Malak and coworkers [13] investigated the addition of shape memory polymer (SMP) to a shape memory alloy (SMA) mesh sheet in order to lock in the actuation obtained from the SMA. This is done by using the heat generated from SMA actuation to soften the SMP, allowing it to deform. When the SMA begins to cool, the SMP drops below its glass transition temperature, solidifying in place and storing the deformation.

This single-field, multimaterial sheet embodies an alternative approach to enhance performance and control to the multifield, multimaterial sheet proposed by Ahmed et al. [12]. In the multifield sheet, MAE and dielectric elastomer (DE) were the proposed active materials, but recent investigations indicate that terpolymer is more effective than DE [7,19] as the second active material due to increased response at lower applied voltages. The first iteration of this multifield sheet is the focus of this paper.

### Origami Modeling.

Origami models are generally of four types: (1) kinematic, (2) analytical, (3) dynamic, or (4) finite element. Each type is useful, and the selection depends on the desired fidelity of the model, the stage in the design process, and input/output requirements. The qualitative properties of each model are summarized in Table 1, and each is briefly discussed next.

Table 1

Comparison between kinematic, analytical, dynamic, and finite element origami models. In general, model complexity and cost increase moving left to right.

KinematicAnalyticalDynamicFinite element
OutputPositionStress and deformationPositionStress and deformation
Rigid panel assumptionYesNoYesNo
Crease modelRevoluteFlexible materialRevolute with torsional spring–damperFlexible material
Solution timeFastFastFastSlow
Complex geometriesYesNoYesYes
KinematicAnalyticalDynamicFinite element
OutputPositionStress and deformationPositionStress and deformation
Rigid panel assumptionYesNoYesNo
Crease modelRevoluteFlexible materialRevolute with torsional spring–damperFlexible material
Solution timeFastFastFastSlow
Complex geometriesYesNoYesYes

Kinematic models are used to determine the motion of a design, or how it folds and unfolds. Closed-form solutions are available using spherical mechanism kinematics [20,21] or screw theory [22]. Simulation-based approaches that utilize optimization are also available [23,24]. In general, the input and output of a kinematic model are solely measures of the position of the origami mechanism. As this is the only objective and panels are considered to be rigid, kinematic models can be solved very quickly.

Analytical models have been developed to account for the interaction of active materials with specific beam geometries. They have been utilized to develop a design concept for a variable curvature electrostrictive polymer beam [25] as well as a model of a multifield terpolymer-MAE bimorph [26]. While very few assumptions are made regarding the crease and active materials models, analytical models are limited to geometries for which analytical kinematic solutions exist.

Dynamic models predict the effect of a load in the form of forces and/or torques on the motion of a design. Like a kinematic model, panels are generally assumed to be rigid. Joints can better approximate creased materials, however, by the addition of torsional spring–dampers, which account for crease stiffness and damping [10]. Dynamic models have been utilized to simulate the effect of magneto-active materials on several origami-inspired designs by approximating them as applied torques.

Finite element models can determine mechanical behavior of the substrate in addition to motion of the model as a result of applied forces. Creases can be simulated directly as flexible material [27], and by using a multiphysics-capable finite element software package, the forces produced by active materials such as SMA, EAP, and MAE can be directly simulated [2731].

While finite element models can theoretically provide a full understanding of a self-folding design, in practice they are generally created for relatively simple folding scenarios or small portions of the overall design. Kinematic models converge to solutions quickly, but no information is obtained regarding the self-folding aspects of the design. Analytical models are versatile in many aspects yet are restricted to simple geometries. A dynamic model is a suitable compromise between these model types, particularly during the development of self-folding designs. The modeling throughout this work is performed with a dynamic model; thus, we assume that the panels are rigid and that joint stiffness and active material effects can be reasonably approximated as described in the following Dynamic Modeling sections.

## Dynamic Modeling: Origami Creases

An origami crease can be viewed as a joint connecting two adjacent panels, which restricts their relative motion to a single rotational axis. This is typically done through the use of revolute joints, which create a mathematical constraint between adjacent panels. They are placed between each panel with each joint's axis of rotation aligned with the shared line between panels. While this approach has been successfully employed in the kinematic modeling of origami, when considering the effects of applied forces and torques (such as those due to applied active materials) on the model's motion, the structural stiffness and damping of the crease must be considered. To account for this, torsional stiffness and damping are added to the revolute joints using torsional spring–dampers. Note that stiffness and damping can be modeled as constants or nonlinear functions. Methods for the determination of torsional stiffness and damping are discussed next.

### Crease Stiffness.

The stiffness of each crease can be calculated by assuming that it acts as a small length flexural pivot (SLFP), a common compliant mechanism joint [32]. The torsional stiffness of an SLFP, K, is given as
$K=EIl$
(1)

where E is the elastic modulus of the material, I is the area moment of inertia of the flexure's cross section, and l is the length of the flexure (see Fig. 2).

Fig. 2
Fig. 2
Close modal

Two methods by which to use Eq. (1) are explored: (1) assume a constant modulus or (2) use a tangent modulus, which is defined here as the relationship between the modulus of a material and the fold angle of the crease. Note that the term “modulus” is used loosely in this discussion because in origami models with large fold angles the material deformations can be well beyond the elastic regime. As such, modulus, for lack of a better term, is used to describe a resistance to deformation that can be measured as a stress–strain relationship. The decision to use a constant or tangent modulus depends on whether or not the modulus as just defined changes with fold angle. In either case, we rely on experimental material characterization to directly determine the modulus. As examples, the stress–strain relationships for three potential crease materials, polydimethylsiloxane (PDMS), polypropylene, and notebook paper, were measured, as shown in Figs. 35, respectively. Stress–strain data were obtained using tensile specimens tested in a dynamic mechanical analysis (DMA) instrument (TA Instruments RSA-G2 Solids Analyzer2). Each material is assumed to be isotropic, though for notebook paper this is a simplification.

Fig. 3
Fig. 3
Close modal

First, note the strain range of these tests, with PDMS (Fig. 3) exhibiting up to 350% strain before failure, polypropylene (Fig. 4) exhibiting over 16% strain (sample not tested to failure), and notebook paper (Fig. 5) failing at a very low strain (approximately 1.5%). The maximum strain expected in a crease can be determined by investigating the beam kinematics of the flexible material. Assuming the panels are rigid, all of the strains are located in the flexible crease (see Fig. 2), which can be calculated as

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal
$ε=−y/ρ$
(2)

where y is the distance from the neutral axis, and ρ is the radius of curvature. The strain is maximum at the outer fibers; thus, y = ±t/2. From the equation for arc length (Eq. (3)), a relationship (see Fig. 6) between flexure length l, radius of curvature ρ, and fold angle, θ, can be written as

Fig. 6
Fig. 6
Close modal
$l=ρθ$
(3)
Equation (3) can be rearranged to solve for ρ, which, when setting y = −t/2 and substituting into Eq. (2), yields Eq. (4) for the maximum strain in the flexible crease as a function of fold angle
$εmax=tθ/2l$
(4)

Considering the flexure lengths, crease material thicknesses, and fold angles investigated in this work, maximum strain is determined using Eq. (4) as approximately 11% (see Table 2). As such, the first 11% of these materials' stress–strain curves are of particular importance for modeling the creases of the origami-inspired designs discussed later in the paper. Although each material type under consideration experiences different levels of strain prior to failure, it is assumed that Eq. (4) applies over their respective strain ranges. In Fig. 7, the first 11% of the three PDMS stress–strain curves from Fig. 3 are shown.

Fig. 7
Fig. 7
Close modal
Table 2

Values used in the calculation of maximum strains (see Eq. (4)) seen for the examples presented later in this work

Paper (barking dog)PDMS (multifield bifold)
t0.070 mm0.412 mm
l1 mm5 mm
$εmax$0.24%10.8%
Paper (barking dog)PDMS (multifield bifold)
t0.070 mm0.412 mm
l1 mm5 mm
$εmax$0.24%10.8%

By taking the slope of the stress–strain curve over the strain range of interest, the elastic modulus of the material is obtained. Though the data in Fig. 7 are quite noisy, it is seen that there is an approximately linear increase in stress for a corresponding increase in strain over the entire range. A linear curve fit was performed with good correlation (R2 = 0.95), yielding a slope, and therefore elastic modulus of 555 kPa for PDMS. As this value is constant over the strain range of interest (see Table 2), Eq. (1) may be used with little loss in accuracy.

For polypropylene and paper, on the other hand, the modulus, or slope of the stress–strain curve, is not constant over the strain range (see Figs. 4 and 5). In this case, the tangent modulus may be used, obtained by taking the numerical derivative of the experimental stress–strain data (see Figs. 8 and 9). Note, in particular, that polypropylene's tangent modulus decreases by an order of magnitude over the strain range of interest. This is a strong indicator that using a constant modulus in Eq. (1) will likely lead to inaccuracy in the stiffness approximation.

Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal
The tangent modulus method is then used by fitting a curve to the tangent modulus–strain relationship, using Eq. (4) to relate the maximum strain to the crease fold angle, and substituting it into Eq. (1). Equation (1) thus becomes Eq. (5), which can be used to define a nonlinear torsional stiffness in the dynamic model
$K=E(εmax(θ))Il=E(θ)Il$
(5)

A current limitation of the tangent modulus method is that it is based on the maximum strain experienced by the crease material rather than the average strain experienced by the crease. This underestimates the modulus, resulting in lower than actual torsional stiffness. The magnitude of the discrepancy is dependent on the particular material in question as well as the determined average strain value. As an example, consider a flexible crease made of polypropylene with a maximum strain of 11% (calculated using Eq. (4)). Utilizing the maximum strain would result in a tangent modulus of approximately 0.2 GPa, obtained from Fig. 8. By making a rough estimate of average strain as half of the maximum, the tangent modulus would instead be determined as 0.35 GPa (again using Fig. 8), a 57% increase over that obtained by using the maximum strain. Developing an equivalent modulus function that takes this average into account is an opportunity for future work.

Neither the constant nor tangent modulus method takes into account the fold history of the specific crease material. For example, polypropylene becomes less stiff the more times it has been folded. Paper is typically creased, which physically damages the material and significantly changes its properties [33,34]. In these cases, both the constant and tangent modulus methods may produce inaccurate results; thus, an experimental method in which the relationship between torsional stiffness and fold angle is measured directly may be more suitable.

### Crease Damping.

In this work, one of the primary outputs of interest from the dynamic model is the set of final folding angles of an origami-inspired design. The addition of structural damping to the dynamic model does not affect the final folding angles, but it does change the time required for the model to reach those angles. For the dynamic simulations performed in this work, structural damping is thus not as critical of a parameter as the crease stiffness. This is not the case for applications where the time required for folding or joint forces during folding are of interest, as the choice of damping can have large effects on these parameters.

In the dynamic model, the structural damping present in an origami-inspired design is accounted for by the addition of torsional dampers to each joint. A simple model of the torsional damping coefficients can be determined by approximating the behavior of creases and their associated panels as second-order rotational systems (see Fig. 10). This simplistic approach is sufficient for this work as each simulation performed is allowed to reach steady state prior to any joint angle measurements. In this model, the damping coefficient, c, for a given torsional damper can be calculated as

Fig. 10
Fig. 10
Close modal
$c=2ζKI$
(6)

where ζ is the damping ratio, K is the crease stiffness, and I is the mass moment of inertia of the panel, which can be approximated as a lumped mass located at its center-of-mass. If more accuracy is required, the rotational mass moment of inertia of each panel about its crease axis can be calculated and used in place of the lumped mass approximation. Note that no values of ζ could be found in the literature for the crease materials used in this work, though for metals it is typically less than 0.01 and for rubber, 0.05 [35]. As the crease materials used here are typically polymers, a value of 0.05 was selected for this work, and more precise values for ζ can be obtained through future experimentation, e.g., by using the logarithmic decrement method [36].

## Dynamic Modeling: Magneto-Active Elastomer

### Material Description and Actuation Method.

Magneto-active elastomers (MAEs) comprise hard-magnetic filler particles, such as barium hexaferrite, embedded in an elastomer matrix [17]. When placed in a magnetic field, the MAE rotates to align its magnetization direction with that of the applied field [37]. When bonded to a passive substrate, this MAE alignment can be used to create bending and folding. MAE has been used in the self-folding of an accordion geometry [29,38] as well as a bistable arch [9,39]. In both of these applications, an elastomer, namely, polydimethylsiloxane (PDMS), was used as the substrate upon which MAE was placed.

MAE is generally fabricated by mixing a two-part elastomer with barium hexaferrite particles. This viscous mixture is used to fill molds of the desired shape, which are then cured in a magnetic field. The poling direction of an MAE is determined during curing, where the magnetic particles align themselves in the direction of the applied field.

The MAE patches discussed in this work interact with an externally applied magnetic field to produce an actuating moment that is an aggregated response of the magnetic particles distributed throughout the patch. The $i$ th individual particle within an MAE patch experiences the Zeeman (external field) energy density
$Ui=−μoMi⋅H$
(7)
where $μ0$ is the relative magnetic permeability of free space, $Mi$ is the magnetization vector of an individual particle, and $Hi$ is the externally applied field local to the particle. The total energy of all the particles in a patch can be given as
$U=−∑iμ0MiHi sin (θi)$
(8)
where $Hi$ is the magnitude of the external field local to the particle, $θi$ is the angle between $i$ th particle's magnetization vector and the local field, and the $i$ th magnetization vector has magnitude $Mi$. When an external field is applied, a particle seeks an energy minimum, exerting a magnetic torque density on the patch whose magnitude can be determined from $Ti=(dUi/dθi)$, which in turn yields
$T=μ0∑iMi×H$
(9)
Assuming uniformity in particle magnetization vectors, the total patch magnetization can be determined from
$M=∑iMi$
(10)
where the resultant patch magnetization, $M$, is in the nominal alignment direction. By similarly assuming uniformity of the external field local to each particle, the familiar relationship found in Eq. (11) is determined. Note that uniformity of the magnetic field also rules out field gradients such that magnetic forces are not present
$T=μ0M×H$
(11)

Equation (11) indicates that the torque produced by an MAE can change in both magnitude and orientation as it moves in relation to the applied field. The magnitude of the torque is largest when the angle between the magnetic moment and applied field is 90 deg, and this magnitude decreases to zero when the magnetic moment and the applied field are aligned. Note that for the models developed in this section, constant magnetization equal to the material's remanence independent of applied field is assumed, and shape effects are ignored. In reality, applied fields of sufficient strength could modify or even reorient a material's remanent magnetization, and the shape of a magnet can affect its interaction with the applied field.

### Actuation Model.

An accurate MAE actuation model should take into account both the changing torque magnitude and orientation as an MAE patch rotates through a magnetic field. To accomplish this in the dynamic model, three parameters defining the magnetic moment vector with respect to the center-of-mass marker of the panel upon which it resides were created for each MAE patch (e.g., m2x, m2y, m2z). Three parameters defining the global orientation of the applied magnetic field were also specified (e.g., Hx, Hy, Hz).

At each time increment in the simulation, a 3-1-3 transformation matrix is used to transform the locally defined magnetic moment vectors to the global coordinate system (e.g., M2x, M2y, M2z). The cross-product is then utilized in the definition of a three-axis torque (e.g., Tx, Ty, Tz) as
$Tx=M2y*Hz−M2z*HyTy=M2z*Hx−M2x*HzTz=M2x*Hy−M2y*Hx$
(12)

## Dynamic Modeling: Electro-Active Polymer

### Material Description and Actuation Method.

Electro-active polymers are a class of materials that exhibit a mechanical response when subjected to an electric field. There are many different types of EAP including dielectric elastomer, electrostrictive polymer, conductive polymer, ionic polymer metallic composites, and liquid crystalline polymer [11,39]. Each of these materials changes shape by way of a unique mechanism, and this shape change can be utilized in devices to produce bending and folding actuators. By attaching the EAP to a passive substrate, the expansion of the active material causes a strain mismatch between the two materials and the result is a bending motion (see Fig. 11).

Fig. 11
Fig. 11
Close modal

In this work, an electrostriction-based relaxor ferroelectric polymer, P(VDF-TrFE-CTFE) terpolymer, is used as the active material for actuation as it has been demonstrated to be effective in actuating origami designs [7]. Its shape changing properties are attributed to the conformational changes of the crystalline phases of polymer from randomly oriented α phase to ordered β phase upon application of an electric field [15]. This causes a thinning and stretching in the material that can result in electrostrictive strains of up to 7% [14,40].

An electric field is applied from a power supply to the EAP by way of wires, which must be connected to compliant electrodes attached to both faces of the EAP. Possible electrode types include carbon grease, sputtered silver, and conductive rubber. Several electrode types have been investigated with the PVDF-based terpolymer, and sputtered silver is used as electrode material in this work as it has been shown to be effective [7].

In order to obtain large deformations at reasonable voltages, thin layers of EAP are often stacked upon one another, resulting in a sandwich of different materials [7]. Neglecting the extremely thin (3 μm) adhesive that binds each layer to one another, we can consider the passive substrate to be connected to the first EAP layer via the electrode. An electrode sits atop each EAP layer, connecting to an additional EAP layer above it. Each electrode must have its own wiring providing power of the proper polarity.

### Actuation Model.

In a dynamic model, the effect of an EAP actuator on the motion of the model must be approximated as a combination of forces and/or torques. When subjected to an applied electric field, an EAP actuator bends with constant curvature [41], and this curvature increases with electric field strength. A free–free beam subject to an applied moment at either end also bends with a constant curvature. This shared behavior can be utilized to approximate the effect of EAP in the dynamic model.

Using the Euler–Bernoulli beam equation [42], the applied moment, M, which causes a specific curvature, κ, is
$M=EIκ$
(13)

where E is Young's modulus, and I is the area moment of inertia of the beam's cross section. By taking the product of the EAP actuator's flexural rigidity, EI, and its curvature at a specific electric field strength, an equivalent moment produced by that actuator can be determined and utilized in the dynamic model. This equivalent moment or torque is then applied to the appropriate panels (the panels upon which the ends of the actuator are affixed), and it is oriented to align with the bending in the actuator.

As the actuator is a multilayer beam (see Fig. 12), the calculation of EI is not as straightforward as it is for a simple prismatic section. For a multilayer multimaterial beam, the flexural rigidity can be calculated as

Fig. 12
Fig. 12
Close modal
$EI=∫−h/2h/2w(y)E(y)y2dy$
(14)

where w(y) is the width of the beam at height y, and E(y) is the elastic modulus of the beam at height y.

In this work, κ is determined by solving a system of equations (see Eq. (15)) for a multilayer EAP that takes into account the material properties and geometry of the EAP, electrode, and adhesive layers [43]. Equation (15) is derived using the Euler–Bernoulli beam equation [44,45] applied to a unimorph actuator with one active layer and one passive layer (see Fig. 11). The multilayer system modeled here replaces the active layer with alternate layers of adhesive, electrodes, and EAP (see Fig. 12). Further details regarding the model derivation are found in Ref. [43]. Each parameter found in Eq. (15) is defined in Table 3 with additional clarification provided in Fig. 12. Note that the heights, h, are simply computed from the thickness of each layer. Note that for the electro-active polymer, Mi = Mp = 3 × 10−18$m2/V2$ and for the adhesive/electrode and the passive substrate Mi = Mg = Ms = 0

Table 3

Definitions of the parameters used in Eq. (15) to determine the curvature of a multilayer EAP actuator

 n Number of layers ES Substrate modulus tS Substrate thickness Ei Modulus of the ith layer ti Thickness of the ith layer hi Height between the top of the ith layer and the top of the substrate (see Fig. 12) ε0 Strain at y = 0 (see Fig. 12) κ Beam curvature Mi Coefficient of electrostriction of the ith layer
 n Number of layers ES Substrate modulus tS Substrate thickness Ei Modulus of the ith layer ti Thickness of the ith layer hi Height between the top of the ith layer and the top of the substrate (see Fig. 12) ε0 Strain at y = 0 (see Fig. 12) κ Beam curvature Mi Coefficient of electrostriction of the ith layer
$(Ests+∑i=1nEi(hi−hi−1)∑i=1nEi2(hi2−hi−12)−Ests22∑i=1nEi2(hi2−hi−12)−Ests22∑i=1nEi3(hi3−hi−13)+Ests33){ε0κ}={∑i=1nEiMi(hi−hi−1)∑i=1nEiMi2(hi2−hi−12)}$
(15)

The calculated values for EI and κ can be used in Eq. (13), yielding an approximation of the moment or torque produced by the multilayer EAP actuator which can then be used in the dynamic model.

## EAP-Actuated Barking Dog

To verify that the developed EAP torque approximation is valid, a previously performed self-folding origami experiment [7] is compared to a developed dynamic model. An origami design known as the barking dog [46] was folded from notebook paper (see Fig. 13), upon which a single four-layer terpolymer actuator (6 mm × 50 mm) was placed. The center fold of the sample was pinned in two places to a thin piece of wood so as to avoid artificially restricting the folding process. The sample was tested at three different frequencies of applied electric field strengths ranging from 20 to 65 MV/m. This applied field caused the jaws to close, and the change in angle of the jaws was determined by digital postprocessing of a video taken during the experiment using ImageJ [47]. A sample frame with the measured angle is found in Fig. 14. Relevant material properties and thicknesses for the sample are found in Table 4, and the experimental data gathered are shown in Fig. 15.

Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal
Fig. 15
Fig. 15
Close modal
Table 4

Relevant parameters of the terpolymer-actuated barking dog experiment

 Epaper 5.5 GPa tpaper 0.070 mm Eterpolymer 0.2 GPa tterpolymer 0.023 mm Eadhesive 0.75 GPa tadhesive 0.0025 mm θinitial 46 deg
 Epaper 5.5 GPa tpaper 0.070 mm Eterpolymer 0.2 GPa tterpolymer 0.023 mm Eadhesive 0.75 GPa tadhesive 0.0025 mm θinitial 46 deg

### Modeling and Validation.

The experimental sample dimensions were measured precisely, and a corresponding dynamic model was created in adams, a commercial multibody dynamics software package (see Fig. 16)3. Following the steps of dynamic model development found in Ref. [10], each panel was created and extruded, after which revolute joints were placed on interconnected panels. The model was created in the flat configuration, after which constant velocity rotational motions were placed on several of the joints to move the model into the initial folded configuration used in the experiment (see Fig. 14). Torsion springs were then applied to each joint so that this initial folded configuration would also be the static equilibrium state of the model. Initial approximations of the joint stiffnesses (using the constant modulus method) were calculated from the material properties and crease geometry (see Table 5) using Eq. (1).

Fig. 16
Fig. 16
Close modal
Table 5

Initial stiffnesses of each of the three crease lengths for the barking dog model found using Eq. (16)

Initial (N mm/deg)
K10.117
K20.363
K30.071
Initial (N mm/deg)
K10.117
K20.363
K30.071
Since there are three distinct crease lengths in this particular model (see Fig. 13) and the stiffness varies proportionally with crease length, the stiffness per millimeter was determined by modifying Eq. (1) by dividing by the crease length, resulting in
$Kmm=Epapertpaper312l$
(16)

where the values for E and t are found in Table 4, and l, the width of the crease, is measured to be approximately 1 mm. This results in a value of 0.00274 N/deg for Kmm, and initial values for K1 and K3 (see Fig. 13) calculated using this parameter are found in Table 5.

The center crease stiffness was calculated to account for the presence of the terpolymer actuator attached directly to the crease (see Fig. 13). Looking at a cross section of the center crease (see Fig. 17), there are three distinct sections, one with the terpolymer actuator and two without. An approximation of the effective stiffness of this crease can be determined by summing the stiffness of each section. The stiffness of each of the substrate-only sections, KS, is the same and can be calculated by multiplying the length of this section by Kmm. The flexural rigidity (EI) of the terpolymer actuator is calculated using Eq. (14), which is then used in Eq. (1) to determine KT. An initial value for K2 can thus be calculated as

Fig. 17
Fig. 17
Close modal
$K2=2KS+KT$
(17)

The effect of the four-layer terpolymer actuator was approximated as two torques placed on the panels containing the ends of the actuator. The torque magnitudes were calculated using Eq. (13) and are listed in Table 6. In Fig. 18, the results of this simulation are compared to the average of the experimental data from Fig. 15.

Fig. 18
Fig. 18
Close modal
Table 6

Torque approximation for a four-layer terpolymer actuator for specified applied field strengths

Field strength (MV/m)Torque (N mm)
200.114
350.349
500.712
651.203
Field strength (MV/m)Torque (N mm)
200.114
350.349
500.712
651.203
It was found that approximating the stiffness using the constant modulus method (Eq. (1)), without any modification in the dynamic model, results in lower than expected changes in jaw angle. This is an indication that the calculated stiffness values used were too high. For this reason, we have tuned the stiffness of the torsional springs in the dynamic model of the barking dog by introducing a crease stiffness correction factor, β, as shown in the below equation
$Kcalibrated=βK$
(18)

The need for the correction factor is likely attributed to the history dependence of the paper substrate, namely, that the crease stiffness of paper is drastically lower after the first time the paper is folded [22]. This is in part due to the fact that when creasing paper, the paper fibers in the crease are damaged the first time it is folded. The current stiffness approximation methods do not take into account the precreased nature of the experimental model. Other potential reasons include the difficulty of determining a precise crease width, l, to be used in Eqs. (1) and (16), and that the assumption that creases act like hinges, i.e., the flexural rigidity of the hinge is much less than that of the panels to which it is connected, is not satisfied since the panels and creases are the same material.

As can be seen in Fig. 18, after applying a correction factor of β = 0.39, the model predicted with acceptable accuracy the results of the experiment, particularly at higher electric field strengths. The modified stiffnesses from Eq. (18) are presented in Table 7.

Table 7

Calibrated stiffnesses of each of the three crease lengths for the barking dog model found using Eq. (18). Note that the correction factor, β, for this model was 0.39.

Calibrated (N mm/deg)
K10.0456
K20.1416
K30.0277
Calibrated (N mm/deg)
K10.0456
K20.1416
K30.0277

From this comparison between the experimental data and the dynamic model of the barking dog, the terpolymer approximation is verified such that we have confidence in using it in the dynamic model of the multifield bifold.

## Multifield Bifold Modeling

### Bifold Design.

The multifield bifold design consists of two active materials, terpolymer and MAE, attached to a polydimethylsiloxane (PDMS) substrate (see Fig. 19). These specific active materials were chosen due to successful actuation of origami-inspired designs in previous experiments [7,9,29]. PDMS was selected for the passive substrate due to its favorable elastic properties and ease of casting into thin sheets.

Fig. 19
Fig. 19
Close modal

The MAE is placed such that it would fold the model along one crease subject to a horizontally applied magnetic field, and the terpolymer is placed such that it would fold the model along the other crease subject to an electric field applied to the electrodes. Note that the terpolymer actuator was fabricated in an “H” shape for several reasons. First, prior experimentation has shown that a high aspect ratio actuator improves fold angles [48]; thus, multiple actuator strips with a high aspect ratio perform better than a single large strip with a lower aspect ratio (given the same actuator area). Each actuator typically requires its own set of wires, but by connecting two actuators via a small electrode segment (i.e., the H shape in Fig. 19), one set of wires can provide power to two actuators, reducing experimental and fabrication complexity.

### Fabrication and Testing.

The specific PDMS sample used during the experiments did not have a perfectly uniform thickness; thus, measurements were taken with a micrometer to determine an average overall sheet thickness, tPDMSavg, as well as average thickness along the two creases, tPDMSmag and tPDMSelec. These thicknesses and those of the active materials, tterpolymer and tMAE, are summarized in Table 8.

Table 8

Substrate and active material thickness. As the PDMS substrate thickness varied, an overall average sheet thickness was computed from micrometer measurements along with average crease thickness.

 tPDMSavg 0.412 mm tPDMSmag 0.462 mm tPDMSelec 0.362 mm tterpolymer 0.026 mm tMAE 0.508 mm
 tPDMSavg 0.412 mm tPDMSmag 0.462 mm tPDMSelec 0.362 mm tterpolymer 0.026 mm tMAE 0.508 mm

The MAE patches were created by using 30% weight v/v 325 mesh barium hexaferrite dispersed in PDMS with dimensions and thickness found in Fig. 19 and Table 8. A measure of the remanence was determined by testing a small sample of the fabricated MAE in a vibrating sample magnetometer, which resulted in a value of 0.000367 A m2 for a patch with dimensions shown in Fig. 19. Each MAE patch was poled out of plane and configured on the PDMS as seen in Fig. 19. The MAE patches were then adhered firmly to the PDMS (see Fig. 20). As a magnetic field is applied from left to right (assuming the same orientation as found in Fig. 19), the MAE will attempt to align with the applied field. This results in the left and right sides rotating out of the page, forming a “V” (see Fig. 21). The sample was tested inside of a large horizontally oriented Walker 7HF electromagnet, and the angle of the sample was measured from digital images taken as the magnetic field strength was varied from 0.0083 to 0.1195 T (see Fig. 21). Three tests over the range of field strengths were performed on the same sample.

Fig. 20
Fig. 20
Close modal
Fig. 21
Fig. 21
Close modal

The terpolymer actuators were created using a single layer of terpolymer with thickness given in Table 8 and dimensions depicted in Fig. 19. To prepare the terpolymer for actuation, the sample was sputtered with a silver electrode, typically 50 nm thick. Two copper wires were attached to each H-shaped terpolymer with one wire on each face of the sample acting as positive and negative leads. After fabrication, adhesive spray (scotch super 77) was used to attach the terpolymer to the PDMS substrate (see Fig. 20). The sample was hung from needle nose tweezers and oriented such that gravity did not oppose the folding, and the wires were connected to a high-voltage power supply (see Fig. 22). A camera was placed under the sample, and video was taken as the applied electric field strength was increased from 30 to 80 MV/m. A sample frame from this video is shown in Fig. 23. The video was postprocessed for graphic analysis, and the change in fold angle was measured using digital image correlation. Two different single-layer actuators were tested on the same substrate a total of three times.

Fig. 22
Fig. 22
Close modal
Fig. 23
Fig. 23
Close modal

### Modeling and Validation.

The experimental bifold sample dimensions were measured precisely, and a corresponding dynamic model was created. The model is composed of four rigid panels connected to one another with four revolute joints. Additionally, one crease on each centerline is attached to ground. Depending on the desired actuation, one of these is active, and the other is inactive. To avoid issues that can arise when simulating folding from the flat state, the creases that are not being actively folded are locked, preventing any revolute motion.

Torsion springs are placed on each hinge, with stiffnesses calculated using the constant modulus method (Eq. (1)) as the modulus of PDMS does not change appreciably over the expected strain range (see Fig. 7). This results in a crease located along the magnetic fold line with stiffness of 0.00126 N mm/deg, and along the electric fold line of 0.00172 N mm/deg. Note that while the magnetic fold line consists of only PDMS in the hinge, the electric fold line also contains the terpolymer actuator, thus increasing the stiffness (see Fig. 19).

Values for the torque magnitude (|M||H|) for specific applied field strengths are listed in Table 9. Since this torque can change in both magnitude and orientation due to the cross-product, three-component torques are placed on each panel (see Fig. 24).

Fig. 24
Fig. 24
Close modal
Table 9

Torque magnitudes (|M||H|) for a single MAE patch at specified magnetic field strengths. Note that torque magnitude and orientation can change as the patch rotates through the field.

Applied field strength (T)Torque magnitude (N mm)
0.00830.030
0.02220.081
0.03610.132
0.05000.184
0.06390.235
0.07780.286
0.09170.337
0.10560.388
0.11950.439
Applied field strength (T)Torque magnitude (N mm)
0.00830.030
0.02220.081
0.03610.132
0.05000.184
0.06390.235
0.07780.286
0.09170.337
0.10560.388
0.11950.439

The torque produced by the single-layer terpolymer actuators was determined using the same process as used for the barking dog actuator with the exception that there are two strips located on each panel. As the location of a torque on a rigid panel has no effect on the motion (assuming the orientation is kept constant), the torque generated by both actuators is combined into one single component torque on each panel (see Fig. 25). For a single-layer terpolymer actuator with the dimensions given in Fig. 19 and Table 8, the torque over the range of field strengths is found in Table 10. Comparisons between the bifold experimental data and model for magnetic and electric actuation are found in Figs. 26 and 27, respectively.

Fig. 25
Fig. 25
Close modal
Fig. 26
Fig. 26
Close modal
Fig. 27
Fig. 27
Close modal
Table 10

Torque magnitudes for two single-layer terpolymer actuators at specified field strengths

Applied field strength (MV/m)Torque (N mm)
300.00654
400.0116
500.0182
600.0262
700.0356
800.0465
Applied field strength (MV/m)Torque (N mm)
300.00654
400.0116
500.0182
600.0262
700.0356
800.0465

In both the magnetic and electric cases, the experiment and simulated results agree well for low field strengths, indicating that the stiffness and active materials approximations are accurate in this regime. At high field strengths, the simulated magnetic fold angles remain close to the experimentally measured angles, but the simulated electric fold angles diverge from the experimental values. The saturation in the experimental results is attributed to self-clearing of the silver-electroded terpolymer. Self-clearing refers to the localized breakdown of dielectric films due to the presence of impurities, such as pinholes or embedded foreign particles. Because of the localized breakdown, some of the metal electrode vaporizes but the EAP continues to operate albeit with a reduced active area [4952]. Due to this smaller area, the electromechanical performance of the EAP is lower, resulting in the model overpredicting the folding angle since the model does not take into account this self-clearing effect. It is noted that the model works well up to electric field magnitudes high enough to successfully actuate most origami-inspired structures, such as the barking dog shown in Figs. 14 and 15. The model thus predicts the fold angle of an ideal terpolymer actuator, and incorporation of self-clearing behavior into the model is a matter for future work.

## Conclusions and Future Work

As a step toward the development of a self-folding magnetic–electric sheet, a dynamic model of self-folding origami was extended to include the effect of electro-active polymer actuators on the motion of an origami design. Details regarding the modeling of the crease stiffness, crease damping, MAE actuation, and EAP actuation were presented. The developed EAP approximation method was validated using experimental data from a barking dog origami model that utilized a four-layer terpolymer actuator. To match the experimental data, the joint stiffnesses were modified using a crease stiffness correction factor, as the current stiffness approximation methods do not take into account the folding history of the substrate.

A design for a bifold multifield sheet was introduced and shown to fold along one axis due to magnetic actuation and along a second axis due to electric actuation. The bifold was fabricated using MAE, single-layer terpolymer actuators, and a passive PDMS substrate. A dynamic model of the bifold was found to accurately predict the experimental fold angles at low field strengths, indicating that the developed active material and crease stiffness approximations are valid for this regime. The MAE model remained close to the experimental results at high field strengths, but the EAP model diverged from experimental results due to self-clearing in the terpolymer actuator.

Having fabricated, modeled, and validated the performance of a multifield bifold sheet, more complicated sheets and folds can be designed. The validation performed in this work gives confidence that, given the correct values for stiffness, a dynamic model can predict the motion of a multifield sheet due to the application of MAE and terpolymer. This enables initial multifield design work to be performed via modeling and simulation, saving time and cost compared to trial-by-error design techniques.

As the current crease stiffness methods do not take into account the folding history of the sample, a matter of future work is a fundamental study experimentally investigating the effects of creasing and folding on crease stiffness for materials that are particularly susceptible to folding history (e.g., polypropylene and paper). In addition, hysteresis loops of the stress–strain behavior can be measured to better understand potential viscous effects. From this experimental data, a more sophisticated crease stiffness approximation integrating the sample's folding history could potentially be determined.

## Acknowledgment

The authors would like to thank Shreya Trivedi and Corey Breznak for assistance in fabricating and characterizing the MAE samples, and Dr. Amira Meddeb for sharing her experimental data regarding the PDMS stress–strain curves used in this work. We gratefully acknowledge the support of the National Science Foundation EFRI Grant No. 1240459 and the Air Force Office of Scientific Research. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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