An Energy-Based Nonlinear Dynamic Model of Dielectric Elastomer Minimum Energy Structures with Stiffeners: Equilibrium Conguration and the Electromechanical Response

The Dielectric Elastomer-based Minimum Energy Structures (DEMES) pertain to an equilibrium conﬁguration attained by the assembly of a pre-stretched electroactive polymer ﬁlm and a compliant boundary frame. Because of their unique characteristics, such as fast response and a large reversible stroke; DEMES have been used widely in the development of soft robotic transducers. However, their utility is typically restricted because of the warping resulting from anticlastic curvature. The present investigation examines the effectiveness of stiffeners in controlling this warping, as well as their effect on the resulting electromechanical response when the DEMES is driven electrically. To this end, we devise an energy-based analytical model that predicts the initial equilibrium conﬁguration of an elementary rectangular DEMES with a ﬁnite number of stiffeners adhered to the boundary frame. The proposed framework uses the neo-Hookean hyperelasticity model for the polymer ﬁlm and the linear elastic constitutive model for both the frame and the stiffeners. Predictive capability of the proposed analytical model is established through com-parisons with 3D ﬁnite element simulations and experimental observations. The analytical model is then extended in the setting of the least-action principle to investigate the complex nonlinear dynamic behavior of the DEMES emanating from the interplay between material and geometric nonlinearities. The proposed dynamic model provides crucial insights into the role of varying levels of geometric and material parameters on the attainment of the initial equilibrium conﬁguration of DEMES and its DC dynamic response when driven by a Heaviside electric load. In particular, we highlight the favorable impact of adding stiffeners in enhancing the stroke of the DEMES and an ampliﬁcation in the attained equilibrium angle with an increasing spacing between the stiffeners. The analytical model and the results reported in this investigation can be of potential use in pre-designing the geometrical and material properties of DEMES for enhancing its electromechanical performance.

investigation can be of potential use in pre-designing the geometrical and material properties of DEMES for enhancing its electromechanical performance.
Keywords Dielectric elastomer minimum energy structure (DEMES) · Stiffeners · Equilibrium configuration · Nonlinear dynamics · Phase portraits 1 Introduction Due to their characteristics of undergoing large reversible areal deformations (more  (Fig. 1 b) shows a considerable reduction in warping when compared with the one 89 without reinforcements (Fig. 1 a). The considered DEMES consists of a compliant 90 frame of initial length L 1 , width W 1 , and thickness h, respectively, with a rectangular 91 hole as shown schematically in Fig. 2. The characterizing dimensions of the rect-92 angular cut section of the compliant frame are length C, width W , and thickness h, 93 respectively. An equally bi-axially pre-stretched membrane (Fig. 2) having the same 94 dimensions (length L 1 , and width W 1 ) as that of the compliant frame is affixed to the 95 frame to develop the proposed minimum energy structure.  The upcoming section presents the static analysis of the proposed DEMES to 106 investigate the attained equilibrium configuration of the structure.   The proposed analytical model of the DEMES is shown schematically in Fig. 3.

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To evaluate the associated strain energy of the stretched membrane, it is assumed that 122 the membrane spans across the two extremities edge of the frame as illustrated in   causes the frame to bend along length C of the rectangular hole. Therefore, the energy 137 required to bend the frame is expressed as in which K b f represents the associated compliant frame bending stiffness which 139 can be defined in terms of mechanical and geometrical parameters of the frame as , where E f represents the Young's modulus of the frame.

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To avoid the warping of the DEMES, the stiffeners are also attached to the surface 142 of the compliant frame (Fig. 3). However, on the downside, it enhances the bending 143 stiffness of the frame by the expression: in which K bs is the bending stiffness of the stiffener, W s represents the width of 145 the stiffener, n is the number of stiffeners, N represents the maximum number of 146 stiffeners, and S denotes the spacing between the consecutive stiffeners, respectively.

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In the present study, we keep the number of stiffeners N = 4 for all the analysis.

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Hence, upon simplification, the Eq. (2) for N = 4 stiffeners is rewritten as The total energy required to bend the compliant frame having stiffeners embedded 150 on the surface is obtained by adding Eq. (1) and Eq. (3) as where λ p1 , λ p2 , and λ p3 denote the developed pre-stretch in X 1 , X 2 , and X 3 di- in which λ 1 , λ 2 , and λ 3 denote the developed true pre-stretches in X 1 , X 2 , and X 3 171 directions, respectively.

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The dimensions of the membrane in the actuated configuration becomes L ×W × 173 t 2 , respectively as shown in Fig. 4 c. From Fig. 3 b, the DE membrane length (L) is 174 expressed in terms of bending angle (θ X1 = θ ) of the frame in the X 1 direction as Similarly, the DE membrane width (W ) is expressed in terms of bending angle 176 (θ X2 ) of the frame in the X 2 direction as The bending of the frame along width direction is negligible, hence invoking the 178 assumption of sin The expression for width (W ) of the membrane 179 (Eq. 8) becomes On inserting the expressions of length L (Eq. (7)), and width W (Eq. (9)) in Eq.

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(6), the developed true stretches in the membrane is expressed as a function of bend-182 ing angle (θ ) of the compliant frame as Using the expressions of λ p1 and λ p2 from Eq. 5 in Eq. 10, the final expressions In this investigation, the generalized neo-Hookean material model Colonnelli in which µ is the shear modulus of the DE membrane. Upon inheriting the ex-196 pressions of λ 1 , λ 2 , and λ 3 from Eq. 11 and substituted in Eq. 12, the expression for 197 the strain energy density of the membrane is drafted as In the present study, an ideal dielectric model is adopted Tang Here, ε represents the membrane permittivity, E signifies the developed nomi- The expression for the total free energy associated with the membrane in the 208 actuated configuration is evaluated by taking the volume integral of the total strain 209 energy density (ψ) from Eq. (15) as The total potential energy of the considered DEMES is expressed as the summa-211 tion of total bending energy of frame with embedded stiffeners (Eq. (4)) and the total 212 free energy of the DE membrane (Eq. (16)) and is written as The static behavior of the DEMES is obtained by equating the first derivative of 216 the total potential energy (U) with bending angle θ equal to zero, i.e., dU dθ = 0.    expressed as a function of three principal stretches for the unit fibers that are initially 276 oriented along the directions of three axes in the coordinate system, λ 1 , λ 2 , and λ 3 , 277 or in terms of the strain invariant I 1 , I 2 , and I 3 as

Constitutive modeling 279
In this study, 3M VHB-4910 acrylic tape is the primary material used as the dielectric 280 elastomer. Due to minimal changes in the volume of rubber-type material, incom-281 pressible nature is usually assumed, i.e., I 3 = 1, and the principal stretch in normal 282 direction becomes λ 3 = 1 λ 1 λ 2 or, physically, λ 3 = t f t i .

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In the present study, the strain energy of 3M VHB-4910 acrylic tape is described

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The flow process technique incorporated to measure the attained equilibrium an-297 gle of the DEMES using ABAQUS is revealed in Fig. 10. First of all, an initial hy-298 perelastic membrane with M3D4 mesh element is considered, as depicted in Fig. 10 299 (a). An equal biaxial pre-stretch is provided to the considered membrane ( Fig. 10(b)).

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The obtained principal stresses for different membrane stretches are considered as an 301 initial condition of the membrane. Next, the prestretched DE membrane having the 302 same dimensions as that of the compliant frame (Fig. 10(c)) is affixed on the frame 303 surface to develop the assembly of the membrane and frame, as shown in Fig. 10(d). is depicted in Fig. 10(f). Finally, the equilibrium angle of the DEMES is obtained 308 by using a commercially available online protector ( Fig. 10(g)). The material prop-309 erties of the elastomeric membrane, inextensible PET frame, and 3D printed PLA 310 reinforcements considered for numerical simulations are indexed in Table 2.  prestretch levels. Figure 11 shows the upper and lower deviations in θ eq obtained 326 experimentally for all considered prestretch levels. From Fig. 11, it is also observed   principle and is expressed as        θ eq of the structure. Finally, the combined influence of bending strips width (b) and 449 spacing (S) between the adjacent stiffeners on the attained equilibrium angle (θ eq ) of 450 the structure is investigated by contour plots (Fig. 16). From Fig. 16, it is inferred 451 that the DEMES exhibits the maximum value of θ eq at a lower level of bending strips and is defined as:   In particular, we investigated the effect of (i) levels of pre-stretch, (ii) width of 507 the bending strip, (iii) spacing between the stiffeners, and (iv) applied voltage on the 508 initial equilibrium angle and the angular stroke resulting from the electromechani-509 cal actuation. The initial equilibrium angle attained by the DEMES increases with increasing levels of pre-stretch on virtue of the fact that increasing amount of strain 511 energy is available for bending the compliant frame. On parallel lines, for a fixed 512 number of stiffeners, the initial equilibrium angle is found to increase with increas-513 ing spacing between the stiffeners; whereas it reduces with increasing frame-width. 514 With electric load used to drive the DEMES actuator, an appreciable diminution in the The associated curved lengths and angles based on the embedded stiffeners on the 536 surface of the compliant frame is obtained geometrically by using Fig. 3 c as , ψ 2 = ψ 1 + θ 3 + θ 4 2 = θ 1 + θ 2 + θ 3 + θ 4 2 ψ 3 = ψ 2 + θ 5 + θ 6 2 = θ 1 + θ 2 + θ 3 + θ 4 + θ 5 + θ 6 2 ψ 4 = ψ 3 + θ 7 + θ 8 2 = θ 1 + θ 2 + θ 3 + θ 4 + θ 5 + θ 6 + θ 7 + θ 8 2 (A.1) in which, 2) The curved lengths associated with the incorporated stiffeners depicts in Fig. 3 c 539 are obtained as To estimate the expression for the total kinetic energy of the DEMES (Eq. (21)), Hence, Upon rearranging, we obtained Similarly, we get Finally, on substituting the expressions from Eqs. (A.8 -A.9) in Eq. (A.4), we Similarly, the expression for velocity V 2 is evaluated as Where, 552 x 2 = x 1 + S cos ψ 2 , y 2 = y 1 + S sin ψ 2 (A.12) On substituting the expressions of ψ 2 from Eq. (A.1) in Eq. (A.12), we get 553 Hence, (A.14) Taking the square of Eq. (A.14), we obtained the following expression.
dθ dt 2 (A.16) Hence, the expression for V 2 using Eq. (A.11) is written as By using the trigonometric identity: sin A sin B + cos A cos B = cos(A − B), we get Similarly, by using the following expressions for V 3 , and V 4 x 3 = x 2 + S cos ψ 3 , y 3 = y 2 + S sin ψ 3 x 4 = x 3 + S cos ψ 4 , y 4 = y 3 + S sin ψ 4

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We get, and, 22) The deduced expressions of velocities V 1 , V 2 , V 3 , and V 4 are substituted in Eq.

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(21) to evaluate the final expression for total kinetic energy of the DEMES.

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Appendix B: Expressions of terms (Z 1 -Z 4 ) 568 The required expressions of the terms (Z 1 -Z 4 ) appeared in the nonlinear dynamic 569 governing equation (Eq. 23) are as follows: