Finite element analysis of electroactive and magnetoactive coupled behaviors in multi-field origami structures

Abstract

Active origami-inspired designs, which incorporate active materials such as electroactive polymers and magnetoactive elastomers into self-folding structures, have shown good promise in engineering applications. In this article, finite element analysis models are developed for several bending and folding configurations that incorporate a combination of active and passive material layers, such as electroactive polymer–actuated unimorph benders based on single and multilayer electroactive polymers, electroactive polymer–actuated notched unimorph folding configurations, and a multi-field actuated bimorph configuration. Constitutive relations are developed for both electrostrictive and magnetoactive materials to model the coupled behaviors explicitly. Shell elements are adopted for their capacity of modeling thin films, relatively low computational cost, and ability to model the intrinsic coupled behaviors in the active materials under consideration. The electrostrictive coefficients are measured and then used as input in the constitutive modeling of the coupled behavior. The magnetization of the magnetoactive elastomer is measured and then used to calculate the magnetic torque as a function of the special orientation, which leads to spatial deformation of the magnetoactive elastomers. The objective of the study is to validate the constitutive models implemented through the finite element analysis method to simulate multi-field coupled behaviors of the active origami configurations. Through quantitative comparisons, simulation results show good agreement with experimental data. By investigating the impact of material selection, location, and geometric parameters, this finite element analysis approach can be used in design of self-folding structures, reducing trial-and-error iterations in experiments.

Keywords

Finite element analysis constitutive modeling electroactive polymer magnetoactive elastomer origami multi-field actuation.

Introduction

The concept of origami folding has been drawing intense interest and has inspired novel applications in many different fields, such as biomedical devices, (Hanks et al., 2016, 2017; Mousavi-Khattat et al., 2015), deformable solar cells (Tang et al., 2014; Zirbel et al., 2013), robotics (Cheng et al., 2017; Mu et al., 2015; Onal et al., 2015), stackable paper batteries (Lee and Choi, 2015), and adaptive manufacturing (Yuan et al., 2017). A number of research studies have focused on applications of self-folding structures (Bani-Hani and Karami, 2015; Del Grosso and Basso, 2010; Neville et al., 2017; Silverberg et al., 2015; Wilcox et al., 2015; Yan et al., 2016), where multiple mechanisms are applied to actuate active folding structures, among which are light-responsive polymers (Liu et al., 2012; Ryu et al., 2012), electroactive polymers (EAPs) (Ahmed and Ounaies, 2016a; Ahmed et al., 2015), magnetoactive elastomers (MAEs) (Bowen et al., 2015, 2016, 2017; Crivaro et al., 2016), and shape memory materials (Firouzeh et al., 2017; Hernandez et al., 2017; Zhakypov et al., 2016).

In this study, two types of active materials, relaxor ferroelectric P(VDF-TrFE-CTFE) terpolymer and MAE, are used to actuate origami-inspired active folding configurations. The poly(vinylidene fluoride) (PVDF)-based terpolymer is converted to a relaxor ferroelectric by introducing the chlorotrifluoroethylene (CTFE) monomer as a defect into the ferroelectric P(VDF-TrFE) copolymer (Huang et al., 2004; Xia et al., 2003). Several properties of the terpolymer enable it to be a good actuator, such as a high electrostrictive strain up to 7%, a relatively high dielectric constant of 50, and a moderate breakdown electric field of 400 MV/m (Bauer et al., 2006; Ahmed and Ounaies, 2016b; Ahmed et al., 2016). MAEs are a class of smart materials fabricated by embedding hard magnetic particles such as barium ferrite into an elastomer matrix. When MAEs are placed in an external magnetic field, the magnetized particles align with the external field, thus generating magnetic torques causing rotation and folding of the structure (Breznak et al., 2016; Sheridan et al., 2014; Paris von Lockette and Sheridan, 2013).

Modeling plays an important role in systematically investigating the folding behaviors and predicting deformations and actuation forces in self-folding origami. For EAP-based actuation, analytical modeling and finite element analysis (FEA) are two common modeling methods. The analytical method applies methods like beam theory and potential energy analysis to calculate stress and deformation of a structure given the applied field(s); however, it is not able to analyze structures with complicated geometries. For example, Hanna et al. (2015) developed analytical models to describe the behavior of the generalized origami waterbomb base (WB) and the split-fold waterbomb base (SFWB). In particular, equations were developed for position analysis, potential energy analysis, and force-deflection behaviors to investigate the impact of initial angles and stiffness of the panels on the final folding angles. Qiu et al. (2016) developed an analytical model to study the reaction force of origami structures when they deform, which improved the mechanism-equivalent approach by treating origami structures as redundantly actuated parallel platforms, and introduced repelling screws to model forces in origami structures for the first time. Qiao et al. (2016) introduced a novel design of an origami-inspired pneumatic solar tracking system and provided an analytical model that established explicit relationships between interior pressures and bending angle.

Erol et al. (2016) developed an analytical model to predict the deformation behavior of an arbitrary bimorph consisting of terpolymer and MAE layers, where the geometry was modeled as a one-dimensional beam that conforms to prescribed bending kinematics and equilibrium of forces and moments throughout. Good agreement with experiments was achieved especially for low field actuation within the linear regime.

Recently, Saad Ahmed et al. (2017b) developed an analytical electromechanical model to study how the curvature of active composite beams varies with different numbers of active terpolymer layers, as well as with the ratios of elastic modulus and thickness between active and inactive layers. The major limitation of this method is that the theory assumes a two-dimensional (2D) deformation; that is, there is a large curvature in the bending direction, but negligible curvature in the orthogonal direction.

For FEA modeling of EAP-based actuation, one approach is to approximate the effect of the applied electric field by applying a pair of compressive surface tractions with same magnitude but opposite directions. McGough et al. (2014) and Zhang et al. (2016) have developed FEA models using a compressive surface traction approach to study actuation performance of dielectric elastomer and PVDF-based terpolymer. The major limitation of this approach is that net forces will occur especially at high deformations because of imbalance of surface areas on the two sides; for thin structures, the net forces will lead to notable deviation from experimental results. Another approach is to develop strain energy functions to model the nonlinear response of EAPs. For example, O’Brien et al. (2009) introduced electrostatic energy density into the strain energy function in ABAQUS to study the curling phenomenon of a dielectric elastomer–based composite beam.

MAE-based actuation has been modeled using rigid multi-body dynamics, in which the structure is treated as rigid panels connected with elastic springs and mechanical torques are applied as input. For example, Bowen et al. have developed dynamic models for the WB (Bowen et al., 2015) and Shafer’s frog tongue (Bowen et al., 2016) and optimized the designs by minimizing the error between actuated and target shapes. The advantage of this method is that it significantly reduces degrees of freedom of the structure and therefore shortens computing time by hours compared to an FEA-based approach. However, dynamic methods do not account for deformation within a panel, which is usually non-trivial, to accurately estimate the deformed shape during actuation.

MAE actuation can also be modeled using the FEA method by applying surface tractions on MAE patches with varying magnitudes as the patches rotate in spatial coordinate system, such as the models developed by Sheridan et al. (2014), Sung et al. (2016), and Zhang et al. (2016). Haldar et al. (2016) developed constitutive relations of magneto-active polymers to combine responses of both magnetic and mechanical systems by introducing Maxwell stress contributions to the free energy function. Then, a finite element formulation was developed through a standard Galerkin approach and was quantitatively verified through a routine driver algorithm.

Compared to models developed in previous studies, in this article, instead of using equivalent surface traction, for example, Maxwell stress, to simulate responses of EAPs and MAEs, direct electromechanical and magneto-mechanical couplings are modeled explicitly in the constitutive equations and are applied in FEA, to better predict the responses of these active materials when external fields are applied. These constitutive relations are then implemented using shell elements in FEA for their capacity of modeling thin films, relatively low computational cost, and ability to model the coupled behaviors in the active materials under consideration. Although the configurations shown in this article mostly fold in one direction, the FEA models account for all three spatial coordinates of the materials and therefore are capable of predicting complex and large three-dimensional (3D) deformations of origami-inspired structures. The constitutive equations of the EAP are modified by introducing the electromechanical coupling coefficients into the original mechanical strain–stress relation. The coupling coefficients in the longitudinal direction are measured using experiments, and transverse ones are determined through an FEA bender model; then, those values are used as input of the EAP model. For MAEs, magnitudes and directions of the magnetic torques generated by MAE patches when placed in an external magnetic field are coupled with spatially deformed orientations of the MAEs. The magnetization of the MAE is determined experimentally and then used to calculate magnetic torque under specified external magnetic field. These body torques contribute to the local stress field within the MAE and thus influence macroscopic strain and deformation. The objective of this study is to validate the constitutive relations implemented by FEA to predict multi-field coupled behaviors of the active origami configurations.

The remainder of the article is organized as follows. The constitutive models of terpolymer-based and MAE-based actuations are introduced, followed by their FEA implementation. Then, the FEA models are developed using COMSOL Multiphysics software for several configurations that incorporate a combination of active and passive material layers, such as EAP-actuated unimorph benders based on the single and multilayer EAPs, EAP-actuated notched unimorph folding configurations, and a multi-field actuated bimorph configuration. FEA results are evaluated both qualitatively and quantitatively: first, the deformed shapes in simulation are compared to the corresponding experiments; then, either folding angles or tip displacements are measured and compared.

Terpolymer-based actuation

Constitutive modeling

In general, the electrostrictive response is modeled as a quadratic coupling between strain and electric field. The expression of strain $ε_{ij}$ in terms of electric field $E_{k}$ , $E_{l}$ , and stress $σ_{kl}$ is shown in equation (1) (Li and Rao, 2004; Newnham et al., 1997)

ε_{ij} = S_{ijkl} σ_{kl} + M_{ijkl} E_{k} E_{l}

(1)

where $S_{ijkl}$ is the elastic compliance tensor and $M_{ijkl}$ represents the electromechanical coupling coefficient tensor. In this study, several simplifications and assumptions are made based on experimental conditions and material symmetries. First, all the materials considered are assumed to be mechanically isotropic. Second, the electric field is always applied through the thickness direction of the active material layers, thus the only non-zero term in the electric field vector is $E_{3}$ , where the 3-direction denotes thickness direction and 1- and 2-directions denote in-plane directions. Third, there is no residual stress when the materials are at rest. Finally, tensors are converted into reduced-symmetry matrices using Voigt notation. Therefore, the constitutive relation, taking these assumptions into consideration, is simplified as shown in equation (2)

ε_{I} = S_{IJ} σ_{J} + M_{IJ} E_{J}^{2}

(2)

For convenience of FEA implementation, the stress vector is expressed explicitly in terms of the strain and coupling effects, as shown in equation (3)

σ_{J} = S_{IJ}^{- 1} (ε_{I} - M_{IJ} E_{J}^{2})

(3)

The expanded form of the equation above is shown in equation (4)

\begin{matrix} [\begin{matrix} σ_{1} \\ σ_{2} \\ σ_{3} \\ σ_{4} \\ σ_{5} \\ σ_{6} \end{matrix}] = [\begin{matrix} λ + 2 μ \\ λ \\ λ \\ 0 \\ 0 \\ 0 \end{matrix} \begin{matrix} λ \\ λ + 2 μ \\ λ \\ 0 \\ 0 \\ 0 \end{matrix} \begin{matrix} λ \\ λ \\ λ + 2 μ \\ 0 \\ 0 \\ 0 \end{matrix} \begin{matrix} 0 \\ 0 \\ 0 \\ 2 μ \\ 0 \\ 0 \end{matrix} \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 2 μ \\ 0 \end{matrix} \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 2 μ \end{matrix}] \\ ([\begin{matrix} ε_{1} \\ ε_{2} \\ ε_{3} \\ ε_{4} \\ ε_{5} \\ ε_{6} \end{matrix}] - [\begin{matrix} M_{11} & M_{12} & M_{13} & 0 & 0 & 0 \\ M_{21} & M_{22} & M_{23} & 0 & 0 & 0 \\ M_{31} & M_{32} & M_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & M_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & M_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & M_{66} \end{matrix}] [\begin{matrix} 0 \\ 0 \\ E_{3}^{2} \\ 0 \\ 0 \\ 0 \end{matrix}]) \end{matrix}

(4)

where the parameters $λ$ and $μ$ are expressed as follows

\begin{matrix} λ = \frac{Y υ}{(1 + υ) (1 - 2 υ)} \\ μ = \frac{Y}{2 (1 + υ)} \end{matrix}

in which Y and $υ$ are elastic modulus and Poisson’s ratio of the terpolymer, respectively. In equation (4), since the only non-zero component in the electric field vector is $E_{3}$ , the effective electrostrictive coefficients in this model which will have impact on the coupling response of the terpolymer are $M_{13}$ , $M_{23}$ , and $M_{33}$ . These coefficients will be determined through experiments, which will be described later in section titled “FEA modeling and verification.”

Due to the compliance of the terpolymer layer, it is not convenient to measure the transverse strains $ε_{11}$ and $ε_{22}$ directly. In this study, the assumption is made that transverse strain in the 1-direction equals to that in 2-direction, and the coefficient k is defined as the ratio between transverse strains and longitudinal strain induced purely by external electric field, namely

k = - \frac{ε_{11} (or ε_{22})}{ε_{33}}

(5)

And equivalently, it holds that

k = - \frac{M_{13} (or M_{23})}{M_{33}}

(6)

FEA implementation

The FEA model is developed using the shell module of the commercial FEA package COMSOL Multiphysics. There are two main reasons to adopt shell elements. One is that shell theory is a good approximation for the deformation of thin structures, in which the thickness is less than about 1/20 of the smallest dimension of their midsurface (Armenakas, 2006). By neglecting the normal and shear stresses in thickness direction, shell elements assume a plane stress condition and do not have nodes through thickness, which significantly decrease the total number of nodes, and thus the degrees of freedom in the model. The other advantage is that it is convenient to modify the constitutive equations of shell elements in COMSOL for terpolymer-based actuation and to apply a body moment as input for the MAE-based actuation, which will be discussed later in section titled “FEA modeling and verification.”

For terpolymer-based actuation, the constitutive equations of the shell elements are modified according to equation (6), in which the electromechanical coupling stress terms are added to the original mechanical stresses, specifically the membrane components of second Piola–Kirchhoff stresses in xx-, yy-, and zz-directions with respect to local coordinate system in COMSOL’s shell module. The modified stresses are shown in equation (7)

\begin{matrix} σ_{1} = (λ + 2 μ) ε_{1} + λ ε_{2} + λ ε_{3} \\ - [(λ + 2 μ) M_{13} + λ (M_{23} + M_{33})] E_{3}^{2} \\ σ_{2} = (λ + 2 μ) ε_{2} + λ ε_{1} + λ ε_{3} \\ - [(λ + 2 μ) M_{23} + λ (M_{13} + M_{33})] E_{3}^{2} \\ σ_{3} = (λ + 2 μ) ε_{3} + λ ε_{2} + λ ε_{1} \\ - [(λ + 2 μ) M_{33} + λ (M_{23} + M_{13})] E_{3}^{2} \end{matrix}

(7)

MAE-based actuation

Constitutive modeling

The second type of active material used in this study is the MAE. To fabricate the MAE patches, 30% by volume 325-mesh barium hexaferrite particles were embedded into a polydimethylsiloxane (PDMS) matrix. Details on the fabrication of the MAE material can be found in the works by Bowen et al. (2016) and Paris von Lockette et al. (2011). When an MAE material is placed in an external magnetic field, it will experience a magnetic torque, which always tries to align the magnetization direction with the external magnetic field direction. The magnetic torque T for a unit volume of particles is determined as follows

T = M \times μ_{0} H

(8)

where $μ_{0}$ is the magnetic permeability of vacuum, M is the remnant magnetization vector of the particles per unit volume, and H is the external magnetic field vector. As the MAE patch rotates, the direction of M remains the same with respect to the surrounding polymer matrix, namely, the material coordinate system, but it changes correspondingly in the global coordinate system. Now let the rotation matrix of the MAE patch be R from global coordinate to material coordinate and let $M_{0}$ be the initial magnetization vector in global coordinate system. Then, the rotated M is as follows

M = R M_{0}

(9)

and the current magnetic torque for a unit volume is as follows

T = M \times μ_{0} H = (R M_{0}) \times μ_{0} H

(10)

An equivalent method to express R is to use the displacement of the normal vectors of the shell elements. In this study, the MAEs are poled in such a way that the magnetization M is normal to the surface. Let the initial normal vector of a MAE patch in global coordinate be

n_{0} = [n_{x 0}, n_{y 0}, n_{z 0}]^{T}

(11)

Since M is always along the normal vector n , then $M_{ini}$ can be expressed as follows

M_{0} = M_{0} n_{0} = M_{0} [n_{x 0}, n_{y 0}, n_{z 0}]^{T}

(12)

where $M_{0}$ is the magnitude of M .

FEA implementation

The previous section provided a rationale for using the shell module of COMSOL for the EAP and MAE actuations. In the shell module of COMSOL, the rotation matrix R can be simplified by taking the advantage of the unit normal vector. When a shell element rotates from its original position as shown in Figure 1, the displacement of the normal vector is interpreted as follows

a = [a_{x}, a_{y}, a_{z}]^{T}

(13)

So the rotated unit normal vector is

n = n_{ini} + a = [\begin{matrix} n_{x 0} + a_{x} \\ n_{y 0} + a_{y} \\ n_{z 0} + a_{z} \end{matrix}]

(14)

Then, the rotated magnetization is

M = M_{0} n = M_{0} [\begin{matrix} n_{x 0} + a_{x} \\ n_{y 0} + a_{y} \\ n_{z 0} + a_{z} \end{matrix}]

(15)

In general, the magnetic field strength can be expressed as follows

H = [\begin{matrix} H_{x 0} \\ H_{y 0} \\ H_{z 0} \end{matrix}]

(16)

Substituting M and H into equation (7), we have

T = M \times μ_{0} H = (M_{0} [\begin{matrix} n_{x 0} + a_{x} \\ n_{y 0} + a_{y} \\ n_{z 0} + a_{z} \end{matrix}]) \times μ_{0} ([\begin{matrix} H_{x 0} \\ H_{y 0} \\ H_{z 0} \end{matrix}])

(17)

The magnitudes of the magnetic torques per unit volume in x-, y-, and z-directions are calculated by equation (17)

\begin{matrix} T_{x} = μ_{0} M_{0} [(n_{y 0} + a_{y}) H_{z 0} - (n_{z 0} + a_{z}) H_{y 0}] \\ T_{y} = μ_{0} M_{0} [- (n_{x 0} + a_{x}) H_{z 0} + (n_{z 0} + a_{z}) H_{x 0}] \\ T_{z} = μ_{0} M_{0} [(n_{x 0} + a_{x}) H_{y 0} - (n_{y 0} + a_{y}) H_{x 0}] \end{matrix}

(18)

In the experiments, the device actuated using MAE is placed between a pair of vertical magnets, experiencing a constant external magnetic field H in the positive z-direction. Therefore, the decomposition of H in the global coordinate system is

H = H_{0} [0, 0, 1]^{T}

(19)

where $H_{0}$ is the magnitude of the field H . Equation (17) is then simplified as in equation (20)

\begin{matrix} T = μ_{0} M \times H = μ_{0} (M_{0} [\begin{matrix} n_{x 0} + a_{x} \\ n_{y 0} + a_{y} \\ n_{z 0} + a_{z} \end{matrix}]) \times (H_{0} [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]) \\ = μ_{0} M_{0} H_{0} [\begin{matrix} n_{y 0} + a_{y} \\ - (n_{x 0} + a_{x}) \\ 0 \end{matrix}] \end{matrix}

(20)

In the shell module of COMSOL, the body torque density in x-, y-, and z-directions is directly applied as an input (with unit $N m / m^{3}$ ); specifically in this study

\begin{matrix} T_{x} = μ_{0} M_{0} H_{0} (n_{y 0} + a_{y}) \\ T_{y} = - μ_{0} M_{0} H_{0} (n_{x 0} + a_{x}) \\ T_{z} = 0 \end{matrix}

(21)

FEA modeling and verification

Terpolymer-based actuation of a unimorph bender

When an electric field is applied, the terpolymer will contract in thickness direction and expand in-plane. If we adhere an inactive layer to the terpolymer, namely, a passive substrate, the in-plane expansion will be constrained, causing bending in the composite. The simplest configuration of such an actuator is a unimorph bender, which in this study is composed of a terpolymer layer and a scotch tape layer adhered together, as shown in Figure 2. Of note here is that, due to the simple geometry of the unimorph bender configuration, there is good reliability and repeatability of experimental results; therefore, this configuration is used to compare modeling and experimental results in order to deduce the value of the ratio k defined in equations (5) and (6). In the experiments, glass slides are used to clamp the top part of the sample, which forms a fixed boundary condition. The geometry and material properties of the unimorph bender are listed in Table 1. The material properties of scotch tape were determined experimentally in a previous study (McGough et al., 2014). The elastic modulus of the terpolymer is measured using a Dynamic Mechanical Analyzer (DMA) of model RSA-G2 by TA Instruments. The value of Poisson’s ratio was obtained from the literature (Roland et al., 2004).

Table 1.

Dimensions and material properties of the unimorph bender.

	Terpolymer	Scotch tape
Length (cm)	3.0	3.0
Width (cm)	2.0	2.0
Thickness (mm)	0.03	0.062
Elastic modulus (GPa)	0.2	1.6
Poisson’s ratio (1)	0.485	0.3
Density (kg/m³)	960	1063

The meshed FEA model is shown in Figure 3. The terpolymer layer and scotch tape layer are defined in two separate shell interfaces which share a single common shell layer in the model, so there is only one layer visible in the FEA model. By default, the shell layer represents the position of the midsurface of the real layer in thickness direction, but the position of midsurface can be relocated by setting a desired “offset.” In this model, the terpolymer is treated as the base of this composite beam of which the offset is set equal to zero. The offset of the passive layer, that is, the scotch tape attached to the terpolymer layer, is set to

d_{scotch 1} = \frac{(t_{ter} + t_{scotch})}{2}

where $t_{ter}$ and $t_{scotch}$ denote the thickness of terpolymer layer and scotch tape layer, respectively. Using the same variable name in the displacement matrix, these two shell interfaces are constrained so that they deform as a union.

To obtain the values of the coupling coefficients, experiments are conducted to measure the longitudinal strain $ε_{33}$ induced by application of electric field where the terpolymer sample is free from mechanical constraint, in other words, $σ_{J}$ is zero. In this situation, the coupling coefficient $M_{33}$ can be calculated as in equations (3) and (4), and the measured $ε_{33}$ and calculated $M_{33}$ are shown in Figure 4 as a function of electric field

M_{33} = \frac{ε_{33}}{E_{3}^{2}}

Deformed shapes of both experimental samples and FEA results are shown for electric field ranging from 0 to 70 MV/m and the case $k = 0.7$ in Figure 5. As electric field increases, the bending curvature of the bender also increases in both experiments and FEA results, and good agreement is achieved between FEA and the experiments. In order to quantitatively compare FEA with experiments, the image analysis software ImageJ is used to measure the bending curvature for both experiments and FEA results, and the results are shown in Figure 6. The error bars on each side of the experimental data points represent 1 standard deviation of five repeated tests with different samples. A parametric sweep study of the coefficient k is conducted to investigate its impact on the bending curvature. The bending curvature increases correspondingly with coefficient k, and from Figure 6 we can see that the cases $k = 0.6$ and $k = 0.7$ match experiments fairly well. To determine the best approximation of k, the normalized squared error between FEA and experiments is adopted as the criterion to evaluate the overall deviation of FEA from the experiments for each k value. As a result, the FEA model with k = 0.7 leads to the lowest normalized error, which is 6.7%. Therefore, k = 0.7 is considered as the best approximation and is used in the remaining models in this article.

Actuation of the unimorph bender using multilayer terpolymer

Researchers have applied multilayer active materials layers in order to increase actuation force and displacement (Carpi et al., 2007; Kovacs et al., 2009). According to a recent study by Ahmed et al. (2017), an analytical model is developed to calculate the blocked force and tip displacement of multilayer actuators based on the relative thicknesses and elastic moduli of the active, substrate, and glue layers. It was shown that as the number of active layer increases, there is a competition between the actuation force and tip displacement because of the increasing of bending stiffness. In this article, FEA models are developed for two-layer, four-layer, and six-layer terpolymer-based unimorph benders to investigate the impact of number of the active layers on the bending performance of the actuator and to help design a multilayer configuration where bending is maximized for particular sets of variables: number of layers, thickness, glue, and so on. The schematic representation of a double-layer terpolymer actuated unimorph bender is shown as Figure 7.

The terpolymer films are adhered to each other using spray glue from 3M Corporation. The glue layer increases the overall bending stiffness, but the elastic modulus and the thickness of the glue layer cannot be easily measured experimentally. A parametric sweep study is conducted to investigate the impact of the glue layer on bending curvature. In the study by Ahmed et al. (2017), the same type of glue is used as in this study, and the thickness and the modulus of the glue layer are estimated as 2.5 µm and 0.75 GPa, respectively. The parametric sweep in this article encompasses these values, where the thickness of the glue layer ranges from 1 to 20 µm and modulus ranges from 0.1 to 2 GPa. The goal is to find the particular set of thickness and modulus of the glue layer that leads to good agreement between FEA and experiments for all three cases, namely, two-layer, four-layer, and six-layer terpolymer-based bender, at the same time. It was found that $t_{glue} = 10 μ m$ and $Y_{glue} = 1 GPa$ are the best estimates of the properties of the glue layer; the bending curvatures are compared between FEA and experiments in Figure 8.

From Figures 6 and 8, it can be observed that as the number of terpolymer layers increases from one to six, the bending curvature of the unimorph bender decreases at a specified electric field, which indicates that the increase in the bending stiffness due to the multilayer terpolymer and glue layers shows stronger influence on the bending performance of the unimorph bender than the influence of increasing field. The drop of bending curvature from two-layer to four-layer is greater than that from four-layer to six-layer because the relative increase in the bending stiffness is higher from two-layer to four-layer, thus the bending curvature is reduced more.

Since good agreement is achieved between FEA and the experiments for the unimorph configuration, the material properties estimated in FEA are assumed to be reasonable enough to predict the impact of the glue layer thickness and modulus on the overall bending performance of the bending actuator. A sensitivity study of the glue layer to the bending curvature is conducted, as shown in Figure 9. It is found that when the thickness of the glue layer is doubled, the bending curvature reduces by 7.9%, 18.0%, and 18.0% for two-layer, four-layer, and six-layer terpolymer-based benders, respectively. When the elastic modulus of the glue layer is doubled, the bending curvature reduces by 1.0%, 13.7%, and 18.6% for two-layer, four-layer, and six-layer terpolymer-based benders, respectively. Based on this sensitivity study, the glue layer overall exerts more influence on the four-layer and six-layer benders than the two-layer terpolymer bender, and thickness has more influence than the elastic modulus on the bending of two- and four-layer terpolymer benders, but they seem to have the same impact on the six-layer bender. It is concluded that a good estimate of the material properties of the glue layer is necessary to accurately predict the deformation of the benders, especially for multilayer terpolymer-based samples.

Terpolymer-based actuation of the single-notch folding unimorph configuration

Active folding can be achieved by having non-uniform thickness along the length direction of the sample, in which case-localized bending occurs at the thinner region, that is, the notch region, resulting in folding behavior. The notched unimorph configuration is inspired by this idea. First, a scotch tape layer of the same size as terpolymer layer is adhered to the terpolymer layer. Then, two smaller scotch tape patches are attached onto the former scotch tape layer symmetrically about the centerline of the sample forming a notch. The schematic representation and fabricated sample are shown in Figure 10. Glass slides are used to clamp the top part of the sample, which forms a fixed boundary condition, as shown in Figure 10(b). The length of the panels is 2.5 cm, and the notch is 1 cm. The geometries and material properties are listed in Table 2.

Table 2.

Dimensions and material properties for the folding unimorph configuration.

	Terpolymer	Scotch tape
Length (cm)	6.0	6.0 (2.5 for panels, 1.0 for notch)
Width (cm)	2.0	2.0
Thickness (mm)	0.03	0.062 (single layer)
Elastic modulus (GPa)	0.2	1.6
Poisson’s ratio (1)	0.485	0.3
Density (kg/m³)	960	1063

Figure 11 shows the ability of the FEA model to simulate folding. In an FEA model of the entire geometry of the single-notch configuration and uniform meshing along the width direction, stress concentration is observed on the edge of the notch region along with anticlastic curvature which prevents folding. To solve these issues, two major modifications are made in the FEA model. First, a symmetric boundary condition is applied along the mid-line of the beam surface to eliminate any numerical error that may lead to asymmetry in the deformation, as shown in Figure 11. Second, a finer mesh is assigned near the edge in order to reduce the stress concentration. As can be seen, the improved model succeeds in predicting the folding response.

Deformed shapes of both the experimental samples and FEA results are shown for electric field ranging from 0 to 60 MV/m in Figure 12. As electric field increases, the two panels of the experimental sample deform with curvature concentrated primarily in the notch region. The actuation is hardly seen from initial position to 20 MV/m and is more pronounced after 30 MV/m. The FEA results follow a similar trend as the two panels deform with an increasing electric field, while the notch region always exhibits much higher curvature than in the panels. These results indicate that the FEA is capable of predicting the folding behavior of the notched unimorph.

In order to quantitatively compare FEA and experiments, the folding angle is adopted as the metric. The folding angle $θ$ is measured as follows: tangent lines are drawn at the two center points of both the root and the tip panels and extended so they intersect; then, the exterior angle of the angle formed between the two panels is measured. Figure 13 shows a schematic representation of how the folding angle is defined in this study. Notice that the folding angle measurement is sensitive to the point selected. We select center point of the panels based on the fact that the panels are stiffer than the notch and therefore could be considered as rigid. The image analysis software ImageJ is used to measure the folding angles for both experiments and FEA results.

The values of folding angle are shown in Figure 14. The error bars in experimental data represent one standard deviation of the data points accounting for seven samples in total. The comparison of folding angles agrees with the qualitative observations of the deformed shape. When the error bars are considered, FEA results agree reasonably well with experiments.

The two thumbnail pictures in Figure 14 indicate a case with large actuation and a case with low actuation, respectively, illustrating the difference between top and the bottom of the error bars. This difference is attributed to differences in fabrication quality, and more details will be explained in the “Discussion” section.

Terpolymer-based actuation of the “finger” configuration

Inspired by the folding of the human finger, the terpolymer-based “finger” configuration is developed in which the two notches imitate the function of finger joints and the three panels of different lengths stand for the three segments of a finger. The schematic representation of the “finger” configuration including dimensions and a fabricated sample are shown in Figure 15.

From Figure 16 we can see that localized deformation, that is, folding, happens within the regions of the two notches in both experiments and FEA. At high electric fields, such as 50 and 55 MV/m, larger folding angles occur at the upper notch than at the lower notch, demonstrating that a longer notch will lead to larger folding angle at a specified field, which agrees with the conclusion from previous study (Zhang et al., 2017). Compared with the single-notch configuration, the notch close to the root in the finger model undergoes similar folding angle, while the notch close to the tip folds as well, realizing a larger deformation and a more complex folded shape. The study of the finger configuration illustrates the potential of FEA modeling to predict behaviors of more complex geometries.

Terpolymer-based actuation of the bimorph configuration

Most smart structures are actuated using a single type of active material, therefore respond to a certain type of external stimulus, and the deformed shape can be controlled only by the magnitude of that stimulus. However, multi-field responsive origami structures, which contain more than one type of actuator material, are capable of folding to multiple shapes depending on not only the magnitude of the stimulus but also the type of stimuli that are applied. The bimorph configuration investigated in this study is a multi-field responsive structure that is composed of a terpolymer layer, a scotch tape layer, and two MAE patches, as shown in Figure 17.

The geometries and material properties are listed in Table 3. When electric field is applied, the structure will fold along the notch formed by the gap between two MAE patches. The terpolymer-based actuation of the bimorph configuration is studied in this section, and the MAE actuation as well as simultaneous actuation will be presented in later sections.

Table 3.

Geometries and material properties in the bimorph configuration.

	MAE	Scotch tape	Terpolymer
Length (mm)	12.5	30.0	30.0
Width (mm)	10.0	10.0	10.0
Thickness (µm)	520.0	62.0	30.0
Density (g/cm³)	1.53	1.063	0.96
Elastic modulus (MPa)	1.45	1600	200.0

MAE: magnetoactive elastomer.

In experiments, the top region of the terpolymer layer and the scotch tape layer are clamped by a pair of glass slides, as in unimorph folding experiments. The upper MAE patch is not clamped; therefore, in the FEA model, we leave a small gap on the top of the bimorph main body, which is 0.7 mm based on measurement from experimental photos. The meshed model and zoomed top gap part are shown in Figure 18.

During the electric actuation experiments, the magnetic field is not applied, while the electric field is applied through the thickness of the terpolymer ranging from 0 to 60 MV/m. The deformed shapes are compared between experiments and FEA results, as shown in Figure 19. It can be seen that the FEA results agree qualitatively with experiments in terms of deformed shapes. Another observation is that the bimorph does not fold as much as the unimorph for the same electric field since the MAE patches are more than 15 times thicker than the terpolymer layer and 9 times as thick as scotch tape. Therefore, the panels are stiffened, and the weight of MAE patches provides a notable restricting force for folding actuation.

Similar to the previous folding unimorph case, the folding angle is used as the comparison metric and is measured using ImageJ. Quantitative comparison between experiments and FEA results is shown in Figure 20. The error bars represent ±1 standard deviation on the data points of six samples in total. Taking the error bars into consideration, the FEA results agree reasonably well with the experiments for electric fields lower than 40 MV/m but overestimate the folding at 50 and 60 MV/m. The thumbnail images in Figure 20 show the scenarios of the largest and least folding at 60 MV/m, which reveals the impact of sample fabrication on the folding performance. In fact, it is observed that most of the samples twist during actuation, which influences the measurement by reducing the folding angle; more on this issue will be explained in the “Discussion” section.

MAE-based actuation of the bimorph configuration

As mentioned above, the bimorph configuration can be actuated using either electric field or magnetic field. For magnetic actuation, the two MAE patches are attached to the scotch tape layer in such a way that the magnetizations M are along the x-axis but in opposite directions, as shown in Figure 21. When the bimorph device is placed in a magnetic field H in the positive z-direction, the two patches will generate magnetic torques that will rotate the top patch counterclockwise and bottom patch clockwise. For FEA modeling, the geometry, boundary conditions, bonding method of the shell layers, and mesh condition are the same as shown in the previous terpolymer actuation section. The difference here is that the input for magnetic actuation is the magnetic torque per unit volume in x-, y-, and z-directions, with magnitudes provided in equation (20). In this study, the experiments are conducted under magnetic field strength $μ_{0} H_{0}$ ranging from 38.5 to 233 mT. The deformed shapes are compared between one set of the experiments and FEA results, as shown in Figure 22. From Figure 22, we can see that in the experiments, the MAE patches rotate as expected, the bottom patch rotating clockwise and the top patch rotating counterclockwise, but not as much. The FEA results capture that trend effectively, and the modeling results compare very well with experiments. The deflection of the top patch from the original position is consistently less than the bottom patch because the very top part of the upper MAE patch is close to the glass slides clamping the terpolymer layer; therefore, the root region is stiffened and deflects less, while the bottom patch rotates more freely. This is also the reason why at a low field strength of 38.5 mT, the rotation of the top patch is hard to observe, but this is not the case for the bottom patch.

Folding angles of the magnetic actuation are measured under each specified field strength for both experiments and FEA results using software ImageJ. The comparison of folding angles is shown in Figure 23. The error bars in the experimental data represent 1 standard deviation on each side of data points for a total of five samples. Figure 23 shows that the FEA results agree well with the average values of experiments, verifying the ability of FEA modeling to quantitatively predict the response of MAE-based actuation. The larger error bars especially at the lower field strengths, that is, 38.5 and 77.5 mT, indicate inconsistency among the five tested samples which may come from residual stress at initial position, twisting deformation caused by the unbalanced mass and glue conditions among the samples, and slight detachment of MAE patches from scotch tape layer. More details will be mentioned in the “Discussion” section.

Simultaneous actuation of the bimorph configuration

Applying electric field and magnetic field simultaneously will create more possibilities in the deformed shapes, thus is an important study for complex folding designs. Experiments have been conducted for the bimorph configuration where the magnetic field strength is kept constant at 38.5 mT (5 A in current), and electric field is increased through the terpolymer layer from 20 to 60 MV/m. The geometry, boundary conditions, bonding method of the shell layers, and mesh conditions are the same as shown in the previous sections. Comparison of the deformed shapes between the FEA results and the experiments is shown in Figure 24.

From Figure 24, we can see that at low electric field, the MAE-based actuation dominates and the panels of the sample rotate similarly as shown in Figure 22 at a magnetic field of 38.5 mT. As the electric field increases, terpolymer-based actuation affects the deformation increasingly and the bimorph sample bends rightward after 50 MV/m. The FEA results follow a similar trend as the experiments. In contrast to the electric actuation, in the simultaneous actuation, the bimorph structure deflects to the right without localized folding in the notch region, which is observed in both the experiments and the FEA results. This is because the tip MAE patch attempts to align with the external magnetic field, thus tends to rotate to the left. The farther it goes to the right, the larger the resisting magnetic torque will become, resulting in anticlastic curvature instead of localized curvature in the notch region.

Instead of folding angle in previous sections, the tip displacement of the bimorph structure is adopted as the comparison metric, since after 50 MV/m, the sample is basically a straight line with folding angle equal to 0. Tip displacements in x- and z-directions are measured and compared as shown in Figures 25 and 26, where the error bars represent ±1 standard deviation of the average for six samples. Generally, the FEA results agree well with average experimental values, but the large error bars indicate inconsistency between the samples. The thumbnail images in Figures 25 and 26 illustrate the greatest and least tip displacement, which again shows the importance of fabrication of the samples. In fact, there is only one sample, which is shown as the lower inset in Figure 25, that deforms in a different manner from all the other five samples, indicating that there is not necessarily another deformation mode, but rather fabrication issue. In addition, when the applied electric field is 50 and 60 MV/m, an anticlastic curvature is observed in both the FEA results and experimental samples, preventing the bimorph from deforming further to the right. This phenomenon will be discussed further in the “Discussion” section.

Anticlastic curvature is observed in the simultaneous actuation of the bimorph in both experiment and FEA when the electric field is equal or above 50 MV/m, as shown in Figure 27. The reason for this phenomenon is that when the applied electric field is high enough, the electric actuation will overcome the magnetic torque in the tip MAE patch and force the MAE patch to the right. In this case, instead of folding to the left due to the magnetic actuation, the magnetic torque curls up the MAE patch in the cross section to attempt to align the magnetization to the external magnetic field, leading to the anticlastic curvature in the bimorph. This curvature also prevents any bending or folding from the electric actuation where the planar expansion of the terpolymer further contributes to this curvature behavior.

Thus, we can conclude that the deformed shape of the bimorph depends not only on the magnitudes of the applied electric and magnetic fields but also on the history of application of these fields, which is observed in experiments as depicted in Figure 28. There are three steps in the experiment. First, the magnetic field is held constant at 38.5 mT, while the electric field increases from 40 to 80 MV/m. In step 2, the electric field E is held constant at 80 MV/m, while the magnetic field increases from 38.5 to 233.6 mT. When H ≤ 194.6 mT, the magnetic actuation is not strong enough to overcome the anticlastic curvature, so the bimorph maintains the straight shape. From H = 194.6 to 233.6 mT, the magnetic actuation becomes strong enough to snap the bimorph through to the folded shape. In step 3, E is constant at 80 MV/m, as in step 2, while H decreases from 233.6 to 38.5 mT. This time, since the bimorph begins in a folded shape, the folding remains as the magnetic field decreases. By comparing the second and third rows in Figure 28, we can see that the history of the applied fields affect how the bimorph deforms, which means that with exactly the same external stimuli, the origami structures may be deployed in different manners.

The FEA model is developed to validate the experimental process stated above, and the FEA results are shown in Figure 29. In this article, geometric nonlinearity is accounted for in all of the FEA models, which refers to the fact that the small deformation assumption is not valid; large deformation is considered in the FEA by including nonlinear products of the coordinate variables and their derivatives. Therefore, the FEA models are able to predict the effect of field application history on the deformed shape. As a result, the simulation shows a very similar trend as the experiments: from H = 194.6 to 233.6 mT, the bimorph snaps through from the straight to the folded shape, and when H decreases from 233.6 to 111.6 mT, the folding remains. Also due to the geometric nonlinearity, anticlastic curvature can occur in simulation, as seen in the experiments, to resist the folding of the bimorph.

Discussion

The objective of the study is to incorporate appropriate constitutive models into finite element models to simulate multi-field coupled behaviors of active origami configurations. Through this study, an overall good agreement, both qualitative and quantitative, is achieved between experimental data and FEA results.

Fabrication quality of the samples is crucial to a successful and convincing comparison between simulation and experimental results. In this study, large error bars were recorded for some of the cases, indicating that there are fabrication inconsistencies for some of the samples. For both the unimorph and the bimorph configurations, the quality of adhesion between the layers of the samples is important to avoid twisting under high field actuation, which will notably inhibit folding quality. For example, the two inset thumbnail photos in Figure 14 show a well-fabricated sample generating larger actuation, and a twisted sample providing less actuation, resulting in a relatively large standard deviation in the experimental data. A similar issue is how well the scotch tape and MAE patches are manually glued to the substrate layer, which not only affects the symmetry or balance of the samples but also contributes to consistency between samples and FEA models, in which all the layers are rigidly bound together.

The twisting of the samples influences measurement in the following two aspects. First, it is difficult to determine the folding angle for a twisted sample. As a result, the measurement method for the twisted samples is different from those for the simulated folded shape. For example, the middle point of the tip edge of the bimorph is adopted as the reference point in folding angle measurement, while in FEA there is a single point at the tip from the side view. Second, some samples twist even before applying any electric field; therefore, there is an initial folding angle for 0 MV/m. The true folding angles of the samples are calculated as the current folding angle minus the initial folding angle.

The coefficient k, which is defined by the ratio between transverse strain and longitudinal strain caused by the electromechanical coupling response, is determined as 0.7 to best match FEA to experiments. The value k = 0.7 is beyond the value of 0.5 for Poisson’s ratio under constant volume assumption. In fact, the coefficient k includes the electromechanical coupling response in addition to the pure mechanical response, so the value 0.7 for k is assumed to be reasonable.

Conclusion

The constitutive models implemented through the FEA method successfully predict the coupled responses of the active materials. To model the PVDF-based terpolymer, the electromechanical coupling tensor is included in the strain–stress relation. To model the MAE-based actuation, the magnetic torque components in x-, y-, and z-directions in the global coordinate are determined and the macroscopic strain and deformation are calculated. The FEA results achieved good agreement with experiments both qualitatively and quantitatively for different configurations, including the bending and folding behavior of the terpolymer-based unimorph, the MAE-based actuation of the bimorph, and the simultaneous multi-field actuation of the bimorph. The success in the FEA modeling of the “finger” configuration, the multilayer terpolymer bender, and the simultaneous actuation of the bimorph shows the potential of the FEA method to predict performances of complicated structures. The material properties of the sprayed glue layers are well approximated in the modeling of the multilayer terpolymer bender. The anticlastic curvature in the simultaneous actuation of the bimorph results in an influence of the field history on the final deformed shape, which is well predicted by FEA modeling. By improving fabrication quality of the samples, better consistency in experiments can be achieved and the constitutive model predictions will be further improved.

After the validation, we conclude that the FEA models developed in this article can predict the behaviors of more complex folding patterns, for example, some of the configurations introduced in our previous publications, including the WB (Bowen et al., 2015), Shafer’s frog tongue (Bowen et al., 2016), and Miura-ori pattern (Cowan and von Lockette, 2017).

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Science Foundation EFRI Grant number 1240459, the Air Force Office of Scientific Research, and the Penn State EFRI-Origami group members. 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.

ORCID iD

Wei Zhang

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