A lumped parameter model for strip-shaped dielectric elastomer membrane transducers with arbitrary aspect ratio

In this section, we recall (with minor modifications) the standard DE free-energy modeling framework proposed by Suo in [27]. We consider a generic DE membrane, depicted in figure 3(a). We denote as L₁, L₂, and L₃ the membrane principal dimensions in the reference configuration. Figure 3(b) shows the same membrane when subjected to an electro-mechanical load, consisting of forces f₁, f₂, and f₃ applied along the membrane principal directions and an electric voltage v applied along axis 3. We denote as l₁, l₂, and l₃ the current principal lengths of the deformed membrane, and as q the total electric charge deposited on the electrodes. This section aims at presenting a general modeling framework for the described class of DE membranes. The model needs to relate the applied forces and voltage to the resulting membrane deformation and charge. For the ease of clarity, we treat in two separate subsections the reversible and irreversible cases.

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3.1.1. Reversible DE.

To begin with, we assume that the DE membrane in figure 3 responds to the applied load in a completely reversible way, i.e. no dissipative phenomena such as viscoelasticity are involved. If this is the case, and under the additional assumption of isothermal process, it can be stated that the time derivative of the system Helmholtz free-energy equals the amount of transferred power. This fact corresponds to the following equation:

$$\begin{equation}\frac{{{{\rm{d}}}\Psi }}{{{\rm{d}}t}} = {f_1}\frac{{{\rm{d}}{l_1}}}{{{\rm{d}}t}} + {f_2}\frac{{{\rm{d}}{l_2}}}{{{\rm{d}}t}} + {f_3}\frac{{{\rm{d}}{l_3}}}{{{\rm{d}}t}} + v\frac{{{\rm{d}}q}}{{{\rm{d}}t}}\end{equation} \tag{ 2 }$$

where is the Helmholtz free-energy of the DE, and the terms on the right-hand side correspond to mechanical powers due by forces f₁, f₂, and f₃, and electrical power due to voltage v.

Next, we define the following normalized quantities: membrane principal stretches λ₁ = l₁/L₁, λ₂ = l₂/L₂, λ₃ = l₃/L₃; true stresses σ₁ = f₁/l₂ l₃, σ₂ = f₂/l₁ l₃, σ₃ = f₃/l₁ l₂; true electric field E = v/l₃; true electrical displacement D = q/l₁ l₂; true Helmholtz free-energy density $\psi$ = $\Psi$ l₁ l₂ l₃. Note that the above definitions must be understood in terms of average quantities, since we are working within a lumped setting.

For simplicity, we assume that the DE electrodes are entirely free to stretch and have a negligible impact on the overall elastic force of the DE. Additionally, as common in DE modeling, we assume the elastomer to be incompressible. Therefore, if we denote by V the material volume, the following holds:

$$\begin{equation}V = {L_1}{L_2}{L_3} = {l_1}{l_2}{l_3}\end{equation} \tag{ 3 }$$

Equation (3) readily implies that

$$\begin{equation}{\lambda _1}{\lambda _2}{\lambda _3} = 1\end{equation} \tag{ 4 }$$

By differentiating (4) over time, we obtain

$$\begin{equation}\frac{\rm{d}}{{{\rm{d}}t}}\left( {{\lambda _1}{\lambda _2}{\lambda _3}} \right) = {\lambda _2}{\lambda _3}\frac{{{\rm{d}}{\lambda _1}}}{{{\rm{d}}t}} + {\lambda _1}{\lambda _3}\frac{{{\rm{d}}{\lambda _2}}}{{{\rm{d}}t}} + {\lambda _1}{\lambda _2}\frac{{{\rm{d}}{\lambda _3}}}{{{\rm{d}}t}} = 0\end{equation} \tag{ 5 }$$

which results into

$$\begin{equation}\frac{{{\rm{d}}{\lambda _3}}}{{{\rm{d}}t}} = - \frac{{{\lambda _3}}}{{{\lambda _1}}}\frac{{{\rm{d}}{\lambda _1}}}{{{\rm{d}}t}} - \frac{{{\lambda _3}}}{{{\lambda _2}}}\frac{{{\rm{d}}{\lambda _2}}}{{{\rm{d}}t}}\end{equation} \tag{ 6 }$$

Equation (6) implies that, for an incompressible DE, we cannot control the three principal stretches in a completely independent way.

After replacing the normalized quantities defined above in (2), dividing each side of the equation by the volume V, and expressing the time derivative of λ₃ as a function of the time derivatives of λ₁ and λ₂ according to (6), the following relationship is obtained:

$$\begin{equation}\begin{gathered} \frac{{{{\rm{d}}}\Psi }}{{{\rm{d}}t}} = \left( {\frac{{{\sigma _1} - {\sigma _3} + ED}}{{{\lambda _1}}}} \right)\frac{{{\rm{d}}{\lambda _1}}}{{{\rm{d}}t}} \hfill \\ \quad \quad \; + \left( {\frac{{{\sigma _2} - {\sigma _3} + ED}}{{{\lambda _2}}}} \right)\frac{{{\rm{d}}{\lambda _2}}}{{{\rm{d}}t}} + E\frac{{{\rm{d}}D}}{{{\rm{d}}t}} \hfill \\ \end{gathered} \end{equation} \tag{ 7 }$$

We can now assume that $\psi$ is a function of the following three independent variables:

$$\begin{equation}\psi = \psi \left( {{\lambda _1},{\lambda _2},D} \right)\end{equation} \tag{ 8 }$$

Equation (8) implies

$$\begin{equation}\frac{{d\psi \left( {{\lambda _1},{\lambda _2},D} \right)}}{{{\rm{d}}t}} = \frac{{\partial \psi }}{{\partial {\lambda _1}}}\frac{{{\rm{d}}{\lambda _1}}}{{{\rm{d}}t}} + \frac{{\partial \psi }}{{\partial {\lambda _2}}}\frac{{{\rm{d}}{\lambda _2}}}{{{\rm{d}}t}} + \frac{{\partial \psi }}{{\partial D}}\frac{{{\rm{d}}D}}{{{\rm{d}}t}}\end{equation} \tag{ 9 }$$

By comparing (7) and (9), we obtain:

$$\begin{equation}\begin{gathered} \hfill \left( {\frac{{{\sigma _1} - {\sigma _3} + ED}}{{{\lambda _1}}} - \frac{{\partial \psi }}{{\partial {\lambda _1}}}} \right)\frac{{{\rm{d}}{\lambda _1}}}{{{\rm{d}}t}}{} {} \;\;\;\;\; \\ \hfill \left( {\frac{{{\sigma _2}\, - \,{\sigma _3}\, + \,ED}}{{{\lambda _2}}} - \frac{{\partial \psi }}{{\partial {\lambda _2}}}} \right)\frac{{{\rm{d}}{\lambda _2}}}{{{\rm{d}}t}}\, + \,\left( {E - \frac{{\partial \psi }}{{\partial D}}} \right)\frac{{{\rm{d}}D}}{{{\rm{d}}t}}\, = \,0 \\ \end{gathered} \end{equation} \tag{ 10 }$$

Since (10) must hold for any system trajectory, we conclude that a suitable set of DE constitutive equations is as follows:

$$\begin{equation}\left\{ \begin{gathered} {\sigma _1} - {\sigma _3} = {\lambda _1}\frac{{\partial \psi }}{{\partial {\lambda _1}}} - ED \hfill \\ {\sigma _2} - {\sigma _3} = {\lambda _2}\frac{{\partial \psi }}{{\partial {\lambda _2}}} - ED \hfill \\ E = \frac{{\partial \psi }}{{\partial D}} \hfill \\ \end{gathered} \right.\end{equation} \tag{ 11 }$$

Equations in (11) describe the constitutive material response in general form. Once a specific function $\psi$ is selected, different types of material behavior can be quantitatively described through the developed model.

3.1.2. Irreversible DE.

The model developed in section 3.1.1 is based on the assumption that dissipative phenomena are negligible. In practice, it is well known that DE material exhibits losses which are mostly due to the viscoelasticity of the elastomer. This section aims at extending the previously developed model by accounting for dissipations, as well.

The major difference between a conservative and a dissipative system is in equation (2), which must be modified as follows:

$$\begin{equation}\frac{{{{\rm{d}}}\Psi }}{{{\rm{d}}t}} \le {f_1}\frac{{{\rm{d}}{l_1}}}{{{\rm{d}}t}} + {f_2}\frac{{{\rm{d}}{l_2}}}{{{\rm{d}}t}} + {f_3}\frac{{{\rm{d}}{l_3}}}{{{\rm{d}}t}} + v\frac{{{\rm{d}}q}}{{{\rm{d}}t}}\end{equation} \tag{ 12 }$$

While (2) implies that all the work done on the DE is stored as free energy, (12) states instead that only part of the work is stored in the membrane while the remaining part is dissipated into heat due to irreversible phenomena (i.e. viscoelasticity). By repeating the steps outlined in section 3.1.1, equation (12) can be reformulated in the following way:

$$\begin{equation}\begin{gathered} \frac{{{{\rm{d}}}\Psi }}{{{\rm{d}}t}} \le \left( {\frac{{{\sigma _1} - {\sigma _3} + ED}}{{{\lambda _1}}}} \right)\frac{{{\rm{d}}{\lambda _1}}}{{{\rm{d}}t}} \hfill \\ \quad \quad \; + \left( {\frac{{{\sigma _2} - {\sigma _3} + ED}}{{{\lambda _2}}}} \right)\frac{{{\rm{d}}{\lambda _2}}}{{{\rm{d}}t}} + E\frac{{{\rm{d}}D}}{{{\rm{d}}t}} \hfill \\ \end{gathered} \end{equation} \tag{ 13 }$$

For the irreversible case, we assume that the free-energy density function depends on the order parameters λ₁, λ₂, and D, as well as on some internal variables ξ_j, j = 1,...,M, i.e.

$$\begin{equation}\psi = \psi \left( {{\lambda _1},{\lambda _2},D,{\xi _1}, \ldots ,{\xi _M}} \right)\end{equation} \tag{ 14 }$$

Differentiation of (14) over time results into the following:

$$\begin{equation}\begin{gathered} \frac{{d\psi \left( {{\lambda _1},{\lambda _2},D,{\xi _1}, \ldots ,{\xi _M}} \right)}}{{{\rm{d}}t}} = \frac{{\partial \psi }}{{\partial {\lambda _1}}}\frac{{{\rm{d}}{\lambda _1}}}{{{\rm{d}}t}} + \frac{{\partial \psi }}{{\partial {\lambda _2}}}\frac{{{\rm{d}}{\lambda _2}}}{{{\rm{d}}t}} + \frac{{\partial \psi }}{{\partial D}}\frac{{{\rm{d}}D}}{{{\rm{d}}t}} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\quad + \sum\limits_{j = 1}^M {\frac{{\partial \psi }}{{\partial {\xi _j}}}\frac{{d{\xi _j}}}{{{\rm{d}}t}}} \hfill \\ \end{gathered} \end{equation} \tag{ 15 }$$

Finally, by replacing (15) in (13), we obtain the following:

$$\begin{equation}\begin{gathered} \left( {\frac{{{\sigma _1} {-} {\sigma _3} {+} ED}}{{{\lambda _1}}} {-} \frac{{\partial \psi }}{{\partial {\lambda _1}}}} \right)\frac{{{\rm{d}}{\lambda _1}}}{{{\rm{d}}t}} {+} \left( {\frac{{{\sigma _2} {-} {\sigma _3} + ED}}{{{\lambda _2}}} {-} \frac{{\partial \psi }}{{\partial {\lambda _2}}}} \right)\frac{{{\rm{d}}{\lambda _2}}}{{{\rm{d}}t}} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \;\;\;{} \;{} \;{} \;{} + \left( {E - \frac{{\partial \psi }}{{\partial D}}} \right)\frac{{{\rm{d}}D}}{{{\rm{d}}t}} - \sum\limits_{j = 1}^M {\frac{{\partial \psi }}{{\partial {\xi _j}}}\frac{{d{\xi _j}}}{{{\rm{d}}t}}} \ge 0 \hfill \\ \end{gathered} \end{equation} \tag{ 16 }$$

In order not to violate the second law of thermodynamics, any constitutive material model needs to satisfy inequality (16) under isothermal conditions. Note that, differently from (10) in which the choice of a material model is somehow straightforward, in (16) we have more flexibility in selecting the constitutive equations. A suitable choice is given by:

$$\begin{equation}\left\{ \begin{gathered} {\sigma _1} - {\sigma _3} = {\lambda _1}\frac{{\partial \psi }}{{\partial {\lambda _1}}} - ED + \sigma _1^d \hfill \\ {\sigma _2} - {\sigma _3} = {\lambda _2}\frac{{\partial \psi }}{{\partial {\lambda _2}}} - ED + \sigma _2^d \hfill \\ E = \frac{{\partial \psi }}{{\partial D}} \hfill \\ \frac{{d{\xi _j}}}{{{\rm{d}}t}} = {\varphi _j},\;\;j = 1, \ldots ,M \hfill \\ \end{gathered} \right.\end{equation} \tag{ 17 }$$

where $\sigma _1^d$, $\sigma _2^d$, and ϕ_j, j = 1,...,M, are arbitrary functions chosen such that the following inequality always holds true:

$$\begin{equation}\frac{{\sigma _1^d}}{{{\lambda _1}}}\frac{{{\rm{d}}{\lambda _1}}}{{{\rm{d}}t}} + \frac{{\sigma _2^d}}{{{\lambda _2}}}\frac{{{\rm{d}}{\lambda _2}}}{{{\rm{d}}t}} - \sum\limits_{j = 1}^M {\frac{{\partial \psi }}{{\partial {\xi _j}}}{\varphi _j}} \ge 0\end{equation} \tag{ 18 }$$

As a final remark, note that the choice $\sigma _1^d$ = 0, $\sigma _2^d$ = 0, and ϕ_j = 0, j = 1,...,M, leads to the dissipation-free model (11).

A sketch of the clamped strip DE in the undeformed and deformed state, with all the relevant quantities, is depicted in figures 4(a) and (b), respectively. Compared to the general membrane introduced in section 3.1, the clamped strip DE is characterized by the following additional assumptions:

• The reference system is chosen in such a way that principal axes 1 and 2 lie on the membrane plane, with axis 1 pointing along the actuation direction, and principal axis 3 denoting the thickness direction;
• Quantity L₁ denotes the distance between the clamps;
• The membrane is mechanically loaded via a tensile force f₁ along axis 1 only, i.e. f₁ ≥ 0 while f₂ = f₃ = 0;
• In order to provide sufficient electrical insulation at the unclamped edges of the membrane, the compliant electrodes do not cover the complete surface area L₁ L₂, but rather an effective area ${L_1}L_2^e$ with $L_2^e$ < $L_2$ .

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A strong requirement for the application of the modeling framework presented in section 3.1 is that the DE membrane undergoes a homogeneous deformation (as in figure 3), so that spatially independent quantities can adequately describe it. Clamped strip DEs, on the other hand, undergo an intrinsically inhomogeneous deformation when actuated. This phenomenon makes the direct application of the previously presented lumped model not possible for the type of DEs under investigation. It is pointed out how the theory of continuum mechanics provides an effective means to tackle the modeling of inhomogeneously deforming bodies [43]. The resulting models, however, are generally expressed in terms of spatially-dependent stress and stretch fields, and thus they turn out to be too complicated for control applications. A lumped model of the strip DE, which accounts for the relationship between v, l₁, and f₁ for different membrane aspect ratios in a control-oriented fashion, is developed in the following.

3.2.1. Reversible DE.

For the sake of clarity, we begin by formulating the strip DE model in the dissipation-free case. To achieve the intended goal without the need for a spatially distributed model, we introduce here the concept of equivalent membrane model. A sketch of such an equivalent membrane is shown in figure 4(c). Given a clamped strip DE as described before, its equivalent membrane is defined in such a way that:

• If no electro-mechanical load is applied to both clamped strip and equivalent membrane, they share the same initial geometry in terms of L₁, L₂, L_2e, L₃;
• The equivalent membrane reacts to an electro-mechanical load by undergoing a homogeneous deformation described by l₁, l₂, $l_2^e$, l₃, i.e. all local effects due to necking and non-uniform electrode placement are modeled in an equivalent lumped fashion;
• DE membrane and electrode always stretch by the same amount along principal direction 2, i.e. l₂/L₂ = $l_2^e$/$L_2^e$ = λ₂;
• If both clamped strip and equivalent membrane are deformed by the same l₁ and subject to the same voltage v, they respond with the same force f₁;
• The above conditions hold for DE membranes with different initial geometry.

In other words, given a clamped strip DE with arbitrary geometry subject to a voltage v and displaced up to l₁, its equivalent membrane model permits to describe its force–displacement–voltage relationship with an equivalent lumped input–output description. We point out that that the equivalent membrane model aims at describing the average behavior of the overall structure (DE + clamps) at system level, rather than the DE material itself. Therefore, all quantities such as stresses and stretches must be interpreted as average measures rather than as true local values. Note that both l₁ and l₂ represent independent degrees of freedom for the equivalent membrane. While the former has a clear meaning by construction, the latter has no direct physical interpretation.

In principle, we can obtain a description of the equivalent membrane by simply setting σ₂ = σ₃ = 0 in (11). However, by doing this, we neglect the fact that the strip DE has electrodes which cover a surface area ${L_1}L_2^e$ < $L_1L_2$. This fact has implications on the constitutive model equations, as discussed in the following. We always consider equation (2) as a starting point. When normalizing the macroscopic quantities into material quantities for the strip DE, everything remains the same as in section 3.1.1 except for the electrical displacement, which is now defined as D = q/${l_1}l_2^e$. By repeating all the steps outlined in section 3.1.1 with this new definition of the electrical displacement, and by assuming that σ₂ = σ₃ = 0 holds true for a strip DE, we obtain the following model:

$$\begin{equation}\left\{ \begin{gathered} {\sigma _1} = {\lambda _1}\frac{{\partial \psi }}{{\partial {\lambda _1}}} - {\alpha _e}ED \hfill \\ 0 = {\lambda _2}\frac{{\partial \psi }}{{\partial {\lambda _2}}} - {\alpha _e}ED \hfill \\ E = \frac{1}{{{\alpha _e}}}\frac{{\partial \psi }}{{\partial D}} \hfill \\ \end{gathered} \right.\end{equation} \tag{ 19 }$$

Coefficient α_e is defined as follows

$$\begin{equation}{\alpha _e} = \frac{{L_2^e}}{{{L_2}}} \lt 1\end{equation} \tag{ 20 }$$

and is a direct consequence of the electrode-free protection layers. Implications of this coefficient on the overall model behavior will be discussed later in this section. Note that quantities λ₁, λ₂, and E are formally defined for the equivalent membrane only. However, they can still be interpreted as a measure of average stretches and electric field for the true strip DE, as well.

Model (19) is quite general, since it treats λ₁ and λ₂ as two independent variables. In membrane DE modeling applications, however, some simplifying assumptions are commonly introduced in order to express all principal stretches as a function of λ₁ only. In this way, the DE can be effectively reduced to a one-dimensional mechanical problem [26, 39]. Such simplifying assumptions are tightly related to the membrane aspect ratio α_a , defined as follows:

$$\begin{equation}{\alpha _a} = \frac{{{L_2}}}{{{L_1}}}\end{equation} \tag{ 21 }$$

In particular, we have:

• Uniaxial response—Holds true if α_a ≪ 1, the small aspect ratio makes the membrane deform isotropically along axes 2 and 3, thus we have λ₂ = λ₃ = 1/λ₁ ^0.5;
• Pure shear response—Holds true if α_a ≫ 1, the large aspect ratio makes the membrane contraction along axis 2 negligible with respect to the contraction along axis 3, thus we have λ₂ = 1 and λ₃ = 1/λ₁.

Note that the above cases only describe two distinctive types of DE strip geometries, corresponding to the limit cases of very large and very small aspect ratio, respectively. However, for strip DEs featuring an aspect ratio α_a ≈ 1, the above simplifying assumptions do not hold, so that λ₁ and λ₂ must be considered as independent quantities. Physically, this condition corresponds to cases in which the necking occurring along direction 2 is not negligible. In contrast, we aim at developing a general modeling framework that generalizes pure shear response, uniaxial response, and any configuration in between, depending on the value of α_a .

To achieve this goal within the framework offered by model (19), the only available degree of freedom is in the choice of the Helmholtz free-energy density function $\psi$. In DE literature, a common choice consists of additively decomposed $\psi$ into a mechanical contribution ${\psi ^m}$, describing hyperelastic free-energy, and an electrostatic contribution ${\psi ^e}$, accounting for the energy stored due to the electric field, i.e.

$$\begin{equation}\psi = {\psi ^m} + {\psi ^e}\end{equation} \tag{ 22 }$$

The electrostatic contribution ${\psi ^e}$ is typically obtained by assuming that the DE behaves as a linear dielectric. In this way, its corresponding electrostatic energy density follows the conventional equation for linear capacitors, i.e.

$$\begin{equation}{\psi ^e} = \frac{1}{V}\left( {\frac{1}{{2C}}{q^2}} \right) = \frac{{{\alpha _e}}}{{2{\varepsilon _0}{\varepsilon _r}}}{D^2}\end{equation} \tag{ 23 }$$

where V is the volume of the elastomer, and C is the DE capacitance, given by the parallel-plate capacitor formula

$$\begin{equation}C = {\varepsilon _0}{\varepsilon _r}\frac{{{l_1}{l_{2e}}}}{{{l_3}}}\end{equation} \tag{ 24 }$$

Multiplicative factor α_e arises in (23) as a consequence of the fact that is, by definition, normalized over the true DE volume l₁ l₂ l₃, while q is distributed on a surface $l_1l_2^e$ only.

Concerning the choice of the mechanical free-energy density ${\psi ^m}$, conventional hyperelastic models such as Neo-Hookean, Mooney-Rivlin, Ogden, Yeoh, or Gent represent by far the most popular solutions adopted in literature [26, 27, 44]. For all such models, ${\psi ^m}$ is expressed as an isotropic function of the principal stretches. In the case of clamped strip DEs, however, isotropy at system level is only observed in the uniaxial case (very small aspect ratio). In case of a generic aspect ratio, instead, the material itself is expected to behave isotropically, while the overall system (DE + clamps) turns out to be anisotropic. This anisotropy is due to the fact that the larger the aspect ratio, the smaller the variation of material stretch along principal direction 2 given the same loading conditions at the boundaries. Since the equivalent membrane model aims at providing a system-level description of the strip, the information on such anisotropy needs to be contained in the mechanical free-energy density $\psi^m$. To account for this fact, we propose a novel type of mechanical free-energy density function which explicitly depend on the membrane aspect ratio α_a , given as follows:

$$\begin{equation}{\psi ^m} = {\psi ^m}\left( {{\alpha _a},{\lambda _1},{\lambda _2}} \right) = {\psi ^{m,i}}\left( {{\lambda _1},{\lambda _2}} \right) + {\psi ^{m,a}}\left( {{\alpha _a},{\lambda _2}} \right)\end{equation} \tag{ 25 }$$

Differently from standard DE models in which $\psi^m$ only describes the elastomer material itself, in this case (25) describes the behavior of the full strip systems. Contribution ${\psi ^{m,i}}$ represents the isotropic part of the mechanical free-energy, which describes the DE material itself, and satisfies:

$$\begin{equation}{\psi ^{m,i}}\left( {{\lambda _1},{\lambda _2}} \right) = {\psi ^{m,i}}\left( {{\lambda _2},{\lambda _1}} \right),\;\;\forall {\lambda _1},{\lambda _2}\end{equation} \tag{ 26 }$$

Note that condition (26) holds true for any of the commonly adopted hyperelastic models (e.g. Heo-Hookean, Mooney-Rivlin, Ogden, Yeoh, Gent). Term ${\psi ^{m,a}}$, on the other hand, describes the anisotropic contribution of the mechanical free-energy, and can be understood as an additional energetic contribution which maps the effects of clamping at a lumped level. It is chosen as a λ₂-dependent barrier function which satisfies the following conditions:

$$\begin{equation}\left\{ \begin{array}{l} {\psi ^{m,a}}\left( {{\alpha _a},1} \right) = 0\quad \quad \quad \forall {\alpha _a} \gt 0\\ {\psi ^{m,a}}\left( {{\alpha _a},{\lambda _2}} \right) \gt 0\quad \quad \forall {\alpha _a} \gt 0,\forall {\lambda _2} \ne 1\\ {\psi ^{m,a}}{\alpha _a},\lambda _2^{^{\prime \prime }} \ge {\psi ^{m,a}}\left( {{\alpha _a},\lambda _2^{^\prime }} \right)\quad \quad \forall {\alpha _a} \gt 0,\forall \lambda _2^{^{\prime \prime }} \ge \lambda _2^{^\prime }\\ {\psi ^{m,a}}\alpha _a^{^{\prime \prime }},{\lambda _2} \ge {\psi ^{m,a}}\left( {\alpha _a^{^\prime },{\lambda _2}} \right)\quad \forall \alpha _a^{^{\prime \prime }} \ge \alpha _a^{^\prime },\forall {\lambda _2}\\ \mathop {\lim }\limits_{{\alpha _a} \to {0^ + }} {\psi ^{m,a}}\left( {{\alpha _a},{\lambda _2}} \right) = 0\quad \quad \forall {\lambda _2}\\ \mathop {\lim }\limits_{{\alpha _a} \to + \infty } {\psi ^{m,a}}\left( {{\alpha _a},{\lambda _2}} \right) = + \infty \;\quad \forall {\lambda _2} \ne 1 \end{array} \right.\end{equation} \tag{ 27 }$$

Roughly speaking, the conditions in (27) mean that the anisotropic free-energy contribution ${\psi ^{m,a}}$ is negligible when the membrane has either no lateral contraction (λ₂ = 1) or a very slim aspect ratio (α_a ≪ 1), and becomes more significant as either the lateral deformation or the aspect ratio increase. The introduction of term ${\psi ^{m,a}}$ allows then to map the complex and inhomogeneous material deformation into a constitutive anisotropy at system-level, which is required to represent the effects of the boundary conditions (i.e. clamping) on the overall input-output DE behavior.

By replacing (22), (23), and (25) in (19), we obtain:

$$\begin{equation}\left\{ \begin{array}{l} {\sigma _1} = {\lambda _1}\frac{{\partial {\psi ^{m,i}}\left( {{\lambda _1},{\lambda _2}} \right)}}{{\partial {\lambda _1}}} - {\alpha _e}{\varepsilon _0}{\varepsilon _r}{E^2}\\ 0 = {\lambda _2}\frac{{\partial {\psi ^{m,i}}\left( {{\lambda _1},{\lambda _2}} \right)}}{{\partial {\lambda _2}}} + {\lambda _2}\frac{{\partial {\psi ^{m,a}}\left( {{\alpha _a},{\lambda _2}} \right)}}{{\partial {\lambda _2}}} - {\alpha _e}{\varepsilon _0}{\varepsilon _r}{E^2}. \end{array} \right.\end{equation} \tag{ 28 }$$

As it can be seen, the linear dielectric assumption given by (23) results into an equivalent Maxwell stress along principal directions 1 and 2 (cf (1)). Differently from lumped DE models commonly encountered in literature, however, in (28) the Maxwell stress is weakened by a factor α_e < 1, since the active electrode area does not cover the full elastomer. This fact implies a lower electro-mechanical actuation capabilities of the strip DE compared to a standard membrane fully covered by electrodes. For this reason, it is important to make $l_2^e$ as close as possible to $l_2$, compatibly with electrical insulation requirements.

To better understand how the anisotropic free-energy affects model (28), we consider the following three cases:

• Case 1: α_a ≪ 1—According to (27), this fact implies that the term depending on $\psi^{m,a}$ is dominated by all the other terms in the second equation of (28), thus we can write:
  $$\begin{equation}\left\{ \begin{array}{l} {\sigma _1} = {\lambda _1}\frac{{\partial {\psi ^{m,i}}\left( {{\lambda _1},{\lambda _2}} \right)}}{{\partial {\lambda _1}}} - {\alpha _e}{\varepsilon _0}{\varepsilon _r}{E^2}\\ 0 \approx {\lambda _2}\frac{{\partial {\psi ^{m,i}}\left( {{\lambda _1},{\lambda _2}} \right)}}{{\partial {\lambda _2}}} - {\alpha _e}{\varepsilon _0}{\varepsilon _r}{E^2} \end{array} \right.\end{equation} \tag{ 29 }$$
  Model (29) describes a mechanically isotropic material behavior, and thus it is representative for a uniaxial response;
• Case 2: α_a ≫ 1—According to (27), this fact implies that the term depending on $\psi^{m,a}$ dominates all the other terms in the second equation of (28), thus we can write:
  $$\begin{equation}\left\{ \begin{array}{l} {\sigma _1} = {\lambda _1}\frac{{\partial {\psi ^{m,i}}\left( {{\lambda _1},{\lambda _2}} \right)}}{{\partial {\lambda _1}}} - {\alpha _e}{\varepsilon _0}{\varepsilon _r}{E^2}\\ 0 \approx {\lambda _2}\frac{{\partial {\psi ^{m,a}}\left( {{\alpha _a},{\lambda _2}} \right)}}{{\partial {\lambda _2}}} \end{array} \right.\end{equation} \tag{ 30 }$$
  From (27), we also see that the physically meaningful solution of the second equation of (30) is simply given by λ₂ = 1. Therefore, we conclude that the resulting model describes a pure shear response;
• Case 3: α_a ≈ 1—According to (27), this implies that the term depending on ${\psi ^{m,a}}$ is expected to be on the same order of magnitude of all the other terms in the second equation of (28). Therefore, we conclude that the resulting model describes an intermediate configuration between pure shear and uniaxial cases, and thus it is suitable to describe inhomogeneous deformations with different degrees of anisotropy.

To sum up, the only degrees of freedom we have in model (28) are represented by isotropic and anisotropic contributions of mechanical free-energy density, denoted as ${\psi ^{m,i}}$ and ${\psi ^{m,a}}$ respectively. The former solely depends on the given material, and can be chosen according to many possible models available from hyperelasticity theory. The latter, on the other hand, depends on both material properties and aspect ratio α_a, and has to be calibrated correctly to describe the level of anisotropy of the membrane deformation.

3.2.2. Irreversible DE.

The theory developed in section 3.2.1 can be easily generalized to include dissipative behaviors as well. In order to do that, we recall that any model for dissipative DE membranes must satisfy equations (17) and (18), for a suitable choice of the Helmholtz free-energy density $\psi$ as well as functions $\sigma _1^d$, $\sigma _2^d$, and ϕ_j, j = 1,...,M. Although the framework provided by (17)–(18) is quite general, in this paper we propose a special class of viscoelastic models, particularly suitable for the class of strip DEs under consideration. For convenience, we first introduce the following partitioning for internal variables ξ_j, j = 1,...,M (c.f. section 3.1.2):

$$\begin{equation}\xi = \left[ {\begin{array}{*{20}{c}} {{\xi _1}} \\ \hline {{\xi _2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\xi _{1,1}}} \\ \vdots \\ {{\xi _{1,{M_1}}}} \\ \hline {{\xi _{2,1}}} \\ \vdots \\ {{\xi _{2,{M_2}}}} \end{array}} \right]\end{equation} \tag{ 31 }$$

where M₁, M₂, and M are freely chosen integers such that M₁ + M₂ = M. Then, a convenient choice for the free-energy density function is as follows:

$$\begin{equation}\psi = {\psi ^m} + {\psi ^e} + {\psi ^v}\end{equation} \tag{ 32 }$$

where ${\psi ^m}$ and ${\psi ^e}$ represent mechanical and electrostatic free-energy contributions defined as in (25) and (23), respectively, while the additional viscoelastic term ${\psi ^v}$ is selected as follows

$$\begin{equation}\begin{gathered} {\psi ^v} = {\psi ^v}\left( {{\lambda _1},{\lambda _2},{\xi _1},{\xi _2}} \right) = \sum\limits_{j = 1}^{{M_1}} {k_{1,j}^v\left[ {{\lambda _1} - {\xi _{1,j}}\left( {1 + \text{log}\,\frac{{{\lambda _1}}}{{{\xi _{1,j}}}}} \right)} \right]} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;\quad + \sum\limits_{k = 1}^{{M_2}} {k_{2,k}^v\left[ {{\lambda _2} {-} {\xi _{2,k}}\left( \!\!{1 {+} \text{log}\,\frac{{{\lambda _2}}}{{{\xi _{2,k}}}}} \right)} \right]} \hfill \\ \end{gathered} \end{equation} \tag{ 33 }$$

while the remaining model function are chosen as:

$$\begin{equation}\sigma _1^d = \eta _{1,0}^v\frac{{{\rm{d}}{\lambda _1}}}{{{\rm{d}}t}}\end{equation} \tag{ 34 }$$

$$\begin{equation}\sigma _2^d = \eta _{2,0}^v\frac{{{\rm{d}}{\lambda _2}}}{{{\rm{d}}t}}\end{equation} \tag{ 35 }$$

$$\begin{equation}{\varphi _j} = \left\{ \begin{gathered} - \frac{{k_{1,j}^v}}{{\eta _{1,j}^v}}{\xi _{1,j}} + \frac{{k_{1,j}^v}}{{\eta _{1,j}^v}}{\lambda _1},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{} j = 1, \ldots ,{M_1} \hfill \\ - \frac{{k_{2,j - {M_1}}^v}}{{\eta _{2,j - {M_1}}^v}}{\xi _{2,j - {M_1}}} + \frac{{k_{2,j - {M_1}}^v}}{{\eta _{2,j - {M_1}}^v}}{\lambda _2},\;\;j = {M_1} + 1, \ldots ,M \hfill \\ \end{gathered} \right.\end{equation} \tag{ 36 }$$

where $k_{1,j}^v$,$k_{2,k}^v$, $\eta _{1,0}^v$, $\eta _{2,0}^v$, $\eta _{1,j}^v$, and $\eta _{2,k}^v$, for j = 1,...,M₁ and k = 1,...,M₂, represent constitutive material parameters. Then, by properly modifying in (17) in order to account for a L $L_1L_2^e$ < $L_1L_2$ (similarly to section 3.2.1), by replacing (23), (25), and (32)–(36) in the obtained equation, and by choosing σ₂ = σ₃ = 0, the following material model is obtained:

$$\begin{equation}\left\{ \begin{array}{l} \frac{{{\rm{d}}{\lambda _2}}}{{{\rm{d}}t}} = - \frac{{{\lambda _2}}}{{\eta _{2,0}^v}}\frac{{\partial {\psi ^{m,i}}\left( {{\lambda _1},{\lambda _2}} \right)}}{{\partial {\lambda _2}}} - \frac{{{\lambda _2}}}{{\eta _{2,0}^v}}\frac{{\partial {\psi ^{m,a}}\left( {{\alpha _a},{\lambda _2}} \right)}}{{\partial {\lambda _2}}}\\ \quad \quad + \frac{{{\alpha _e}{\varepsilon _0}{\varepsilon _r}}}{{\eta _{2,0}^v}}{E^2} - \sum\limits_{k = 1}^{{M_2}} {\frac{{k_{2,k}^v}}{{\eta _{2,0}^v}}\left( {{\lambda _2} - {\xi _{2,k}}} \right)} \\ \frac{{d{\xi _{1,j}}}}{{{\rm{d}}t}} = - \frac{{k_{1,j}^v}}{{\eta _{1,j}^v}}{\xi _{1,j}} + \frac{{k_{1,j}^v}}{{\eta _{1,j}^v}}{\lambda _1},\;\;j = 1, \ldots ,{M_1}\\ \frac{{d{\xi _{2,k}}}}{{{\rm{d}}t}} = - \frac{{k_{2,k}^v}}{{\eta _{2,k}^v}}{\xi _{2,k}} + \frac{{k_{2,k}^v}}{{\eta _{2,k}^v}}{\lambda _2},\;\;k = 1, \ldots ,{M_2}\\ {\sigma _1} = {\lambda _1}\frac{{\partial {\psi ^{m,i}}\left( {{\lambda _1},{\lambda _2}} \right)}}{{\partial {\lambda _1}}} - {\alpha _e}{\varepsilon _0}{\varepsilon _r}{E^2}\\ \quad \quad + \eta _{1,0}^v\frac{{{\rm{d}}{\lambda _1}}}{{{\rm{d}}t}} + \sum\limits_{j = 1}^{{M_1}} {k_{1,j}^v\left( {{\lambda _1} - {\xi _{1,j}}} \right)} . \end{array} \right.\end{equation} \tag{ 37 }$$

Note that electric field E and stretch λ₁ (as well as its time derivative) represent the inputs of model (37), while stress σ₁ is the system output, and stretch λ₂ as well as internal variables ${\xi _{1,j}}$, ${\xi _{2,k}}$ define the system state. In case one wants to express the model input and output in terms of measurable quantities f₁, l₁, v rather than material quantities σ₁, λ₁, E, the following transformation equations can be used:

$$\begin{equation}{f_1} = \frac{{{L_2}{L_3}}}{{{\lambda _1}}}{\sigma _1}\end{equation} \tag{ 38 }$$

$$\begin{equation}{\lambda _1} = \frac{{{l_1}}}{{{L_1}}}\end{equation} \tag{ 39 }$$

$$\begin{equation}E = \frac{{{\lambda _1}{\lambda _2}}}{{{L_3}}}v\end{equation} \tag{ 40 }$$

It can also be verified that the thermodynamic consistency condition (18) holds true for any admissible trajectory of (37) (this follows by a straightforward extension of the result presented in [42], details are omitted for conciseness). Moreover, if M₁ = M₂ = M = 0 and $\eta _{1,0}^v$ = $\eta _{2,0}^v$ = 0 then model (37) degenerates in the non-dissipative version given by (28).

Even though model (37) contains a significant number of variables and parameters, some facts can be exploited to gain a better understanding of the different terms. In fact, by inspecting the structure of (37), it can be noted that the corresponding equations are equivalent to the ones describing a pair of nonlinear and mutually coupled spring-damper systems. A graphical depiction of this mechanical equivalence is reported in figure 5. Each one of these two models consists of a parallel connection between a nonlinear spring describing the hyperelastic stress contribution, an electric field-dependent spring representing the electro-mechanical coupling, and a combination of linear springs and dampers describing the rate-dependent viscoelastic effects. Parameters $k_{1,j}^v$, $k_{2,k}^v$, $\eta _{1,0}^v$, $\eta _{2,0}^v$, $\eta _{1,j}^v$, and $\eta _{2,k}^v$ can then be naturally interpreted as stiffness and damping coefficients of the corresponding linear springs and dampers, with M₁ and M₂ denoting the number of spring-damper elements appearing in each spring-damper model, while internal variables ${\xi _{1,j}}$ and ${\xi _{2,k}}$ can be understood as the stretches associated to dampers $\eta _{1,j}^v$ and $\eta _{2,k}^v$, respectively. Although the physical significance of such a graphical interpretation is of limited use, it allows for a clearer interpretation of the role of each term in determining the material response, as well as for a simpler parameter calibration. We also point out that the free-energy based viscoelastic model proposed in this paper introduces rate-dependency by means of additional linear spring and damper elements (i.e. the ones described by coefficients $k_{1,j}^v$, $k_{2,k}^v$, $\eta _{1,0}^v$, $\eta _{2,0}^v$, $\eta _{1,j}^v$, and $\eta _{2,k}^v$), while other authors introduced the same effects via non-linear springs and dampers [27, 30]. This partially linear structure makes the model potentially more suitable for automatic parameter calibration and control applications (e.g. energy-based control [42], LMI-based control [10]), compared to models having a full nonlinear description of the viscoelastic dynamics. As a final remark, note that (37) requires the time derivative of λ₁ as a further input. This may be undesirable in some applications, since such a signal could be affected by measurement noise in case it comes from experimental data. If this is the case, one can set $\eta _{1,0}^v$ = 0 and try to fit the material response by using all the remaining parameters. It is, however, not recommended to set to zero $\eta _{2,0}^v$ as well. Indeed, choosing $\eta _{2,0}^v$ > 0 permits to transform the nonlinear algebraic equation related to λ₂ (i.e. the second one in (28)) into an ordinary differential equation (ODE), which can be effectively integrated by available numerical solvers.

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For the numerical investigation conducted in this section, the isotropic and anisotropic components of the DE mechanical free-energy density function are chosen according to a Yeoh model and a parabolic term, respectively:

$$\begin{equation}{\psi ^{m,i}}\left( {{\lambda _1},{\lambda _2}} \right) = \sum\limits_{i = 1}^3 {{C_{i0}}{{\left( {\lambda _1^2 + \lambda _2^2 + {\lambda _1}^{ - 2}{\lambda _2}^{ - 2} - 3} \right)}^i}} \end{equation} \tag{ 41 }$$

$$\begin{equation}{\psi ^{m,a}}\left( {{\alpha _a},{\lambda _2}} \right) = \beta \left( {{\alpha _a}} \right){\left( {{\lambda _2} - 1} \right)^2}\end{equation} \tag{ 42 }$$

where C_{i 0}, i = 1,2,3, represent constitutive material parameters while β depends on both material and aspect ratio. If β = 0, the material response is uniaxial. The larger β, the more the material behavior will resemble an ideal pure shear response. Since the focus of this first study is on the quasi-static behavior, viscoelastic contributions will be neglected.

Numerical implementation of the lumped model is performed in MATLAB. The FE model, on the other hand, is implemented in COMSOL environment. The FE implementation considers the material as isotropic, i.e. the DE mechanical free-energy ${\psi ^m}$ only accounts for the isotropic part ${\psi ^{m,i}}$, according to (41). The clamping is implemented through properly defined boundary conditions. As a result of the clamping, the membrane reacts to a displacement with an inhomogeneous deformation field.

Different sets of simulations are performed, by considering three different Yeoh models, labeled as Yeoh 1, Yeoh 2, and Yeoh 3, respectively. The coefficients of the three Yeoh models are chosen arbitrarily, according to the following criteria: the stress-stretch curve is within the typical range of values for silicone-based DEs (i.e. stress of 1 to 2MPa for stretches of about 1.5 to 2); each curve has a different shape (i.e. quasi-convex for Yeoh 1, quasi-linear for Yeoh 2, and quasi-concave for Yeoh 3). In this way, we are able to evaluate the accuracy of the proposed modeling framework for a number of possible application scenarios, which reflect the typical order of magnitude of stress and stretch typically encountered for DEs. For each Yeoh model, seven membranes with different aspect ratios values are taken into account, ranging from 0.04 (quasi-uniaxial) to 10 (quasi-pure shear). The numerical values of the adopted parameters are reported in table 1. For each material model and membrane geometry, two displacement-controlled tensile experiments are simulated in COMSOL by considering a maximum stretch λ_1,max = 2 and constant voltage values of v = 0kV and v = 2.5kV, respectively. An exemplary plot of the COMSOL simulation results is shown in figure 6. In here, the material parameters are chosen according to Yeoh model 1, for the strips with aspect ratio α_a = 0.2 (figure 6(a)), α_a = 1 (figure 6(b)), and α_a = 5 (figure 6(c)). The spatial distribution of λ₂ is shown for both initial (left-hand side) and displaced (right-hand side) configurations, by considering v = 0kV and an average λ₁ = λ_1,max = 2 in each case. It can be observed how the case with α_a = 1 shows the highest amount of inhomogeneities.

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The overall FE simulation results are presented as solid lines in figure 7 for Yeoh model 1, in figure 8 for Yeoh model 2, and in figure 9 for Yeoh model 3, respectively. Note that, among all of the conducted simulations, only a restricted set of most meaningful ones is reported, corresponding to α_a equals to 0.04, 0.2, 1, 5, and 10. For each case, results are presented in terms of average stress σ₁ vs. average stretch λ₁ (upper row), as well as average stretches λ₂, λ₃ vs. average stretch λ₁ (lower row). For the FE model, the average stress σ₁ is computed by calculating the reaction force in COMSOL due to the applied deformation, and dividing it by the corresponding cross-sectional area. Average stretch λ₁ is simply obtained according to (39), while average stretch λ₂ is computed according to the following:

$$\begin{equation}{\lambda _2} \equiv \frac{{{\rm{mean}}\left( {{l_2}} \right)}}{{{L_2}}} = \frac{1}{{{L_2}}}\left( {\frac{a}{{{l_1}}}} \right).\end{equation} \tag{ 43 }$$

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Quantity a in (43) is the total surface area of the deformed membrane in the current configuration, and is computed by COMSOL on the basis of the FE simulations results. Note that (43) can be interpreted as a measure of the average stretching along principal direction 2 over the membrane profile. Finally, we compute λ₃ = 1/λ₁ λ₂.

After obtaining the FE simulation results, the same tests are performed by means of the developed lumped model. As stated before, the lumped model makes use of the same free-energy function of the FE one, with the addition of the anisotropic barrier function defined by (42). All the system and material parameters are kept the same for both models, so that the only degree of freedom we have with the lumped model is represented by β appearing in (42). For each Yeoh model and aspect ratio, a specific value of β is computed in such a way to match the results of the FE model. The resulting values of β are reported in table 1. The corresponding simulation results for the lumped DE model are reported as dashed lines in figures 7–9. It can be observed how, by properly tuning the barrier function parameter, the DE simulations match the FE results in a very accurate way for each one of the considered case studies. It is interesting to notice how the agreement is not only observed for the stress response (σ₁ vs. λ₁ curves), but also for the membrane geometric state (λ₂, λ₃ vs. λ₁ curves). A small mismatch in stretches is observed in some cases, e.g. for Yeoh models 1 and 3 with intermediate aspect ratio values, while the membranes with extreme values of aspect ratios are in general reproduced with a high accuracy. This can be explained by inspecting figure 6. On the one hand, the cases with very small (figure 6(a)) and very large (figure 6(c)) aspect ratios exhibit a negligible local strain distribution, this they can be well described by means of a lumped model. On the other hand, in the case of intermediate aspect ratios, the effects of the local strain distribution are significantly more pronounced (figure 6(b)), and thus they are more difficult to describe via averaged quantities by means of a unique parameter β. Despite the deviations observed between FE and lumped models, it is worth noting how these are still very small in magnitude. In particular, it can be noted that the overall trends can be described with remarkable accuracy in each one of the examined scenarios. Based on the above comparison, we can conclude that principal stretch λ₂ appearing in the lumped model can be effectively interpreted as an average measure of lateral stretch along the membrane necking profile. Note also that for α_a = 0.04 and v = 0kV the stretches λ₂ and λ₃ are very close to each other, for all of the considered Yeoh models, as expected from a uniaxial experiment. At the same time, for the case in which α_a = 10 we have that λ₂ is almost constant and equal to 1 independently on the considered Yeoh model and applied voltage, thus well approximating a pure shear type of loading. For all the other aspect ratios, intermediate configurations between uniaxial and pure shear are observed. Note in particular how, for α_a close to 1, neither uniaxial nor pure shear assumptions can be assumed to hold consistently, thus justifying the adopted modeling approach.

To better understand the role of the barrier function in shaping the material response, figure 10 shows how the σ₁ vs. λ₁ curves change as β varies, for different Yeoh models and voltage values. The difference in the shape of the three adopted Yeoh models characteristics can be readily appreciated from this plot. In here, the curves are also directly compared with the exact characteristics obtained for uniaxial (dotted black lines) and pure shear (dashed black lines) response, obtained as the limit cases for infinitely small and infinitely large β, respectively. It can be noted that increasing β makes the material curves gradually transition from the uniaxial (lowest stress) to the pure shear (highest stress) case. The proposed model represents then an effective tool to describe both loading situations, as well as every possible configuration in between, in a control-oriented fashion. Note also that the curves corresponding to v = 0kV tend to move upward when transitioning from uniaxial to pure shear cases, while the corresponding curves for v = 2.5 kV are almost insensitive to the aspect ratio. As a result, the gap between the low-voltage and high-voltage characteristics becomes wider for membranes with larger aspect ratio. Since a larger stress gap facilitates the design of biasing mechanisms for DEA systems (see [2] for details), we conclude that pure shear membranes might be preferable in actuator applications.

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From the simulation study conducted in section 4.1, it can be concluded that the developed model represents an effective way to account for the behavior of DE membranes with different aspect ratio in a control-oriented fashion. One issue with the developed model, however, is related to the choice of the anisotropic mechanical free-energy density component ${\psi ^{m,a}}$. In fact, while all the other parameters can be easily related to physical or geometrical quantities, the ones describing ${\psi ^{m,a}}$ have no physical interpretation, and thus they can only be determined via model fitting based on more accurate numerical studies. This fact would significantly restrict the range of applicability of the proposed model.

To overcome this issue, in this section we perform a numerical study aimed at understanding the relationship existing between the aspect ratio α_a and the parameters describing the isotropic ${\psi ^{m,i}}$ and anisotropic ${\psi ^{m,a}}$contributions of the mechanical free-energy. More specifically, we aim at finding some invariant descriptors, denoted as μ_i , which only depend on the aspect ratio and not on the material model. For the specific class of Yeoh models and parabolic barrier functions considered in (41) and (42), this is equivalent to

$$\begin{equation}\begin{array}{*{20}{r}} {\begin{array}{*{20}{r}} {{\mu _i}\left( {C_{10}^{{\rm{Yeoh}}\;j},C_{20}^{{\rm{Yeoh}}\;j},C_{30}^{{\rm{Yeoh}}\;j},{\beta ^{{\rm{Yeoh}}\;j}}\left( {{\alpha _a}} \right)} \right) = {\mu _i}\left( {{\alpha _a}} \right)} \\ \!\! {\forall {\alpha _a},j = 1,2,3} \end{array}} \end{array}\end{equation} \tag{ 44 }$$

After a suitable descriptor μ_i is obtained, relationship (44) can be inverted and used to express β as a function of the aspect ratio α_a and the given Yeoh parameters C₁₀, C₂₀, C₃₀ related to the isotropic free-energy part. In addition, we arbitrarily define μ_i in such a way it satisfies the following criteria:

$$\begin{equation}\left\{ \begin{array}{l} {\mu _i}\left( {a_a^{^{\prime \prime }}} \right) \ge {\mu _i}\left( {\alpha _a^{^\prime }} \right),\;\;\forall \alpha _a^{^{\prime \prime }} \ge \alpha _a^{^\prime }.\\ \mathop {\lim }\limits_{{\alpha _a} \to {0^ + }} {\mu _i}\left( {{\alpha _a}} \right) = 0\\ \mathop {\lim }\limits_{{\alpha _a} \to + \infty } {\mu _i}\left( {{\alpha _a}} \right) = 1 \end{array} \right.\end{equation} \tag{ 45 }$$

Roughly speaking, μ_i defines how close we are to a pure shear state of deformation. In this way, we have that μ_i = 0 corresponds to uniaxial, μ_i = 1 to pure shear, while any value in between defines how close the membrane deformation is to the corresponding limit cases.

Based on the above definition, in the following we propose eight candidate descriptors:

• Descriptor μ₁: defines how close the actual mechanical stress component $\sigma _1^m$ is to the corresponding stress for uniaxial case, measured in terms of normalized area:
  $$\begin{equation}{\mu _1} = \dfrac{{ {\int_1^{{\lambda _{1,max}}} {\left| {\sigma _1^m - \sigma _1^{m,uniaxial}} \right|d{\lambda _1}} } }}{{ {\int_1^{{\lambda _{1,max}}} {\left| {\sigma _1^{m,pureshear} - \sigma _1^{m,uniaxial}} \right|d{\lambda _1}} } }}\end{equation} \tag{ 46 }$$

• Descriptor μ₂: similar to μ₁, but in here the closeness is measured on the λ₂ vs. λ₁ curve, rather than on the $\sigma _1^m$ vs. λ₁ one, while still considering no electrical actuation:
  $$\begin{equation}{\mu _2} = \dfrac{{ {\int_1^{{\lambda _{1,max}}} {\left| {{\lambda _2} - \lambda _2^{uniaxial}} \right|{{d}}{\lambda _1}} }}}{{ {\int_1^{{\lambda _{1,max}}} {\left| {\lambda _2^{pureshear} - \lambda _2^{uniaxial}} \right|{{d}}{\lambda _1}} } }}\end{equation} \tag{ 47 }$$

• Descriptor μ₃: it is obtained by simply measuring the relative weight between the leading coefficient of the Yeoh model C₁₀ and the barrier parameter β, by exploiting the fact that both of them can be physically related to a stiffness:
  $$\begin{equation}{\mu _3} = 1 - \frac{{{C_{10}}}}{{{C_{10}} + \beta }}\end{equation} \tag{ 48 }$$

• Descriptor μ₄: since β explicitly appears in the equilibrium equation of σ₂ only, it is possible to construct a descriptor by accounting for how much the isotropic component $\sigma _2^{m,i}$ contributes to $\sigma _2^m$, where
  $$\begin{equation}\sigma _2^{m,i}\left( {{\lambda _1},{\lambda _2}} \right) = {\lambda _2}\frac{{\partial {\psi ^{m,i}}\left( {{\lambda _1},{\lambda _2}} \right)}}{{\partial {\lambda _2}}}\end{equation} \tag{ 49 }$$
  Note that β can be related to a stiffness term, which physically corresponds to the derivative of σ₂ with respect to λ₂. Consequently, we measure μ₄ in terms of the magnitude of the slopes of $\sigma _2^{m,i}$ and $\sigma _2^m$ with respect to λ₂, evaluated for a nominal condition corresponding to an undeformed membrane:
  $$\begin{equation}{\left. {{\mu _4} = 1 - \frac{{\left| {\frac{{\partial \sigma _2^{m,i}}}{{\partial {\lambda _2}}}} \right|}}{{\left| {\frac{{\partial \sigma _2^m}}{{\partial {\lambda _2}}}} \right|}}} \right|_{\left( {{\lambda _1},{\lambda _2}} \right) = \left( {1,1} \right)}}\end{equation} \tag{ 50 }$$
• Descriptor μ₅: similar to μ₄, but in here the relative weight between the slopes of $\sigma _2^{m,i}$ and $\sigma _2^m$ is measured in terms of the norm of the total gradient, rather than just the partial derivative with respect to λ₂:
  $$\begin{equation}{\left. {{\mu _5} = 1 - \frac{{\left\| {\left[ {\begin{array}{*{20}{c}} {\frac{{\partial \sigma _2^{m,i}}}{{\partial {\lambda _1}}}}&{\frac{{\partial \sigma _2^{m,i}}}{{\partial {\lambda _2}}}} \end{array}} \right]} \right\|}}{{\left\| {\left[ {\begin{array}{*{20}{c}} {\frac{{\partial \sigma _2^m}}{{\partial {\lambda _1}}}}&{\frac{{\partial \sigma _2^m}}{{\partial {\lambda _2}}}} \end{array}} \right]} \right\|}}} \right|_{\left( {{\lambda _1},{\lambda _2}} \right) = \left( {1,1} \right)}}\end{equation} \tag{ 51 }$$

• Descriptor μ₆: while all the descriptors introduced up to now are constructed via a stress or stretch argument, μ₆ is instead based on energy considerations. More specifically, μ₆ evaluates the relative weight between isotropic and total contributions of the mechanical free-energy density, measured in terms of Hessian matrix determinants for the nominal undeformed state:
  $$\begin{equation}{\left. {{\mu _6} = 1 - \frac{{\left| {\mathcal{H}\left\{ {{\psi ^{m,i}}} \right\}} \right|}}{{\left| {\mathcal{H}\left\{ {{\psi ^m}} \right\}} \right|}}} \right|_{\left( {{\lambda _1},{\lambda _2}} \right) = \left( {1,1} \right)}}\end{equation} \tag{ 52 }$$
  where the Hessian operator of $\psi$ is defined as
  $$\begin{equation}\mathcal{H}\left\{ \psi \right\} = \left[ {\begin{array}{*{20}{c}} {\frac{{{\partial ^2}\psi }}{{\partial \lambda _1^2}}}&{\frac{{{\partial ^2}\psi }}{{\partial {\lambda _1}\partial {\lambda _2}}}} \\ {\frac{{{\partial ^2}\psi }}{{\partial {\lambda _2}\partial {\lambda _1}}}}&{\frac{{{\partial ^2}\psi }}{{\partial \lambda _2^2}}} \end{array}} \right]\end{equation} \tag{ 53 }$$
  The choice behind the Hessian matrix is due to the fact that it depends on second derivatives of the free-energy, i.e. it is related to a stiffness information;
• Descriptor μ₇: similar to μ₆, but in here the relative weight between Hessian matrices is measured in terms of matrix norm (i.e. the induced 2-norm):
  $$\begin{equation}{\left. {{\mu _7} = 1 - \frac{{\left\| {\mathcal{H}\left\{ {{\psi ^{m,i}}} \right\}} \right\|}}{{\left\| {\mathcal{H}\left\{ {{\psi ^m}} \right\}} \right\|}}} \right|_{\left( {{\lambda _1},{\lambda _2}} \right) = \left( {1,1} \right)}}\end{equation} \tag{ 54 }$$

• Descriptor μ₈: similar to μ₆, but in here the relative weight between Hessian matrices is measured in terms of matrix trace:
  $$\begin{equation}{\left. {{\mu _8} = 1 - \frac{{\left| {{\rm{Tr}}\left\{ {\mathcal{H}\left\{ {{\psi ^{m,i}}} \right\}} \right\}} \right|}}{{\left| {{\rm{Tr}}\left\{ {\mathcal{H}\left\{ {{\psi ^m}} \right\}} \right\}} \right|}}} \right|_{\left( {{\lambda _1},{\lambda _2}} \right) = \left( {1,1} \right)}}\end{equation} \tag{ 55 }$$

These eight descriptors are different possible representations for the level of closeness to the pure shear deformation case. Note that μ₁ and μ₂ represent the most accurate measures, since they are directly based on actual material curves, but they might be complex to compute analytically and also explicitly depend on maximal stretch λ_1,max. Conversely, descriptors μ₃–μ₈ represent less accurate ways to quantify the desired behavior, but are much simpler to compute.

Descriptors μ₁–μ₈ are computed based on the results of section 4.1, and reported in figure 11. In particular, each subplot of figure 11 displays a different descriptor obtained for all of the three Yeoh models, by considering the aspect ratio on the abscissa axis. In each plot, the average of the three Yeoh model curves is also reported, depicted as a solid black line. Clearly, the more the curves are packed together, the better index μ_i in describing an exact invariant quantity for the given class of models. It can readily be observed how μ₁ fails in this regard, while μ₂ is consistent among all of the considered models. Therefore, measuring the closeness to pure shear based on stretch λ₂ seems to represent a more coherent way than using mechanical stress $\sigma _1^m$. Concerning the other indexes, mixed results can be observed. In terms of variability, the most consistent one among all the simulation turns out to be μ₃, which is also the simplest one to compute. The problem of such descriptor, however, is that it is defined for a simple class of barrier functions and material models, and thus it cannot be easily generalized as the other ones. The second best descriptor turns out to be μ₆, i.e. the one based on the determinant of the Hessian matrices. This quantity has the advantage of being defined for a generic material. Moreover, an analytical solution is available for the considered class of Yeoh models and quadratic barrier function, given as follows:

$$\begin{equation}{\mu _6}\left( {{C_{10}},\beta } \right) = 1 - \frac{{3{C_{10}}}}{{3{C_{10}} + \beta }}\end{equation} \tag{ 56 }$$

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A direct comparison among all the average curves in figure 11 is also reported in figure 12. In here, it can readily be seen that μ₁ and μ₂ are, on average, very close to each other. At the same time, μ₆ represents a remarkably accurate approximation of both those indexes, while μ₃ significantly deviates from them. All the other descriptors show different levels of agreement to μ₁ and μ₂. In summary, we have that that μ₂ is the most physically accurate and consistent descriptor, but it is complex to compute. On the other hand, μ₆ provides a very accurate approximation of μ₂ and, also, is available in closed form. Therefore, we deduce that index μ₆ represents the most suitable choice for quantifying the effects which α_a has on the level of anisotropy of the material response.

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Based on the above discussion, a simple heuristic rule can be developed to estimate β based on the given Yeoh model and membrane aspect ratio. First, we point out that a good approximation of the μ₆ vs. α_a curve shown in figure 12 is given by the following fitting function:

$$\begin{equation}{\mu _6}\left( {{\alpha _a}} \right) = \frac{{\alpha _a^2 + 1.351{\alpha _a}}}{{\alpha _a^2 + 1.732{\alpha _a} + 4.050}}\end{equation} \tag{ 57 }$$

By equating (56) and (57) and solving for β, we obtain the following useful relationship:

$$\begin{equation}\beta \left( {{C_{10}},{\alpha _a}} \right) = 3{C_{10}}\frac{{\alpha _a^2 + 1.351{\alpha _a}}}{{0.381{\alpha _a} + 4.050}}\end{equation} \tag{ 58 }$$

Equation (58) can be used as a first-order approximation to estimate the barrier function, thus allowing to construct a lumped model without requiring a FE calibration campaign. It is worth noting that (58) has a heuristic nature and might not be valid in general. However, it turned out to be useful at least for the class of models considered in this work.

A picture of the experimental setup used to test the strip DE membranes is shown in figure 13. It consists of a test-bench divided into two parts, one fixed to the ground and the other one capable of performing a displacement on a fixed linear guide. The clamps holding the DE specimen are assembled on the top of these parts, located at the same height from the ground. Therefore, the strip samples act like a planar bridge which connects the linear moving guide and the fixed constrained end of the test-bench. The moving part is actuated by the high-performance AEROTECH direct-drive linear motor ACT 165DL, which is capable of executing prescribed displacement history with maximum amplitude of 200 mm and accuracy of ±3 μm. The electro-mechanical stress response of the specimens is measured with a FUTEK load cell mounted in between the DE clamps and the moving part side of the bench. The electrical connections for the applied voltage on the DEs are situated on the bench fixed part. Dedicated custom-designed clamps hold the membrane strip in a planar fixed position, and ensure at the same time the electrical contact between the electrodes and two spring pre-loaded cupper contacts, one on the HV side and one on the ground (GND) side of the specimen, respectively. A fixed Logitech C920 camera monitors the testbench displacement workspace during the experimental activity, in order to catch the electrode area deformation during the stretching of the DE. All the measured signals, i.e. camera and load cell, as well as the input displacement history, i.e. the actuation of the linear motor, are synchronously acquired and controlled by a National Instrument FPGA board for real-time control and measurements. The experiment output is the measured force of the electro-mechanical response of the DE, together with a synchronized video of the test. The camera captures the moving membrane at a rate of 30 fps, in sync with the time of the load cell acquisitions. Finally, each video frame is elaborated with the MATLAB computer vision toolbox, for a single fixed-camera image processing. This procedure relies on a previous camera calibration test conducted before the experimental activity, in order to estimate the camera intrinsic, extrinsic, and lens distortion parameters. This calibration consisted of 20–40 camera pictures, already fixed on the test bench, of a checkboard with prescribed dimensions in the workspace of the experimental activity, with output the camera parameters. These are then adopted for the video frames processing from the experiments, correcting the distortions and allowing the conversion from pixel to real-world measurements, i.e. mm, for the electrode pixel areas identified through a blob analysis from each picture frame.

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The first set of experiments is conducted by considering clamped DE strips with different aspect ratios. Five different membranes are considered, all of them characterized by the same values of L₂, $L_2^e$, and L₃, while L₁ changes such that the aspect ratio ranges from 0.24 (quasi-uniaxial) to 6 (quasi-pure shear). The exact geometry values are reported in table 2, while a picture of the different membranes is shown in figure 14. All the DE strip specimens are manufactured on the same batch of silicone material made of Wacker ELASTOSIL 2030 with a thickness of 50 μm. The membrane is held from a metallic frame, shown in the upper part of figure 14. Nine samples are manufactured on a single batch through a screen printing process, i.e. one strip specimen with an aspect ratio of 0.24 and two specimens for each other samples. For each one of the five different membranes, two tensile tests are performed by deforming the DE mechanically at 0.1 Hz up to a stretch λ_1,max = 1.75, by considering two different constant voltage values v = 0kV and v = 2.5kV, respectively. The resulting force f₁ vs. λ₁ curves are shown as solid black lines in figure 15, in the upper row for v = 0kV and in the lower row for v = 2.5kV, respectively. Note that in here, differently from section 4, the results are reported in terms of force rather than stress, since this quantity represents the true measured variable. The abscissa axis is still expressed in terms of λ₁, though, for a better comparison among different geometries (note that L₁ differs for each strip). Given the same stretch, an increase in force for higher aspect ratios can be observed, consistently with the results of section 4. A second set of experiments is shown in figure 16, in which the DE force is measured by keeping its deformation at constant λ₁ = 1.5 and by cycling v from 0 to 2.5 kV with a frequency of 0.2 Hz. Also in here, the results are shown for all the tested strip DEs. As a result of the Maxwell stress, the blocking force decreases for increasing voltage.

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Table 2. DE membrane parameters, experimental study.

System parameter	Value	Unit
L1	{4, 10, 20, 40, 100}	(mm)
L2	24	(mm)
L2 e	20	(mm)
L3	50	(μm)
ε0	8.854 × 10−12	(F/m)
Material parameter	Value	Unit
εr	2.8	(–)
C10	2.41 × 105	(Pa)
C20	−3.32 × 104	(Pa)
C30	2.01 × 104	(Pa)
β	According to (58)	(Pa)
M1	1	(–)
M2	0	(–)
k1,1 v	1.31 × 105	(Pa)
η1,1 v	7.93 × 104	(Pas)
η1,0 v	0	(Pas)
η2,0 v	4.02 × 104	(Pas)

Based on the collected results, the DE membrane model parameters are calibrated. Also in this case, the mechanical free-energy density contributions are selected according to (41) and (42), while the viscoelastic model is chosen to fit the measured hysteresis. DE permittivity is selected as ε_r = 2.8, according to the material manufacturer data [41], while the barrier function parameter β is chosen according to the heuristic model (58). In this way, the only free parameters are represented by C₁₀, C₂₀, C₃₀, M₁, M₂, $k_{1,j}^v$, $k_{2,k}^v$, $\eta _{1,0}^v$, $\eta _{2,0}^v$, $\eta _{1,j}^v$, and $\eta _{2,k}^v$. The calibration of those parameters is performed via nonlinear optimization tools available in MATLAB, based on a subset of the experiments reported in figure 15 (corresponding to α_a = 0.6 and α_a = 2.4, respectively). The complexity of the viscoelastic model is kept minimal by priorly setting M₁ = 1 and M₂ = 0. The optimal parameter values are reported in table 2.

The obtained lumped model results are represented as dashed red lines in figures 15 and 16. A remarkably good accuracy can readily be observed. Note that, in this particular case, no FE simulations are performed to assist the calibration of the anisotropic barrier function, but rather its determination relies entirely on the results of section 4.2. We then conclude that the obtained invariant descriptor μ₆ is well capable of accounting for the intrinsic effects of clamping for a wider class of Yeoh models than the calibration ones. It can also be observed that the effects of the Maxwell stress, i.e. the change of the curves with applied voltage, are well described without performing an explicit calibration of ε_r . As a result of this predictive feature, we conclude that our modeling assumptions on the effects that α_e has on the electro-mechanical actuation turn our to be physically consistent. To quantify the accuracy of the obtained simulations, a FIT_% index is computed for each result, according to the following:

$$\begin{equation}FI{T_\% } = 100\left( {1 - \frac{{\left\| {{f_1} - {{\hat f}_1}} \right\|}}{{\left\| {{f_1} - {\rm{mean}}\left( {{f_1}} \right)} \right\|}}} \right)\end{equation} \tag{ 59 }$$

where f₁ and ${\hat f_1}$ represent arrays of measured and simulated force, respectively. Resulting FIT_% values for all experiments are reported in table 3. As it can be seen, the FIT_% is always larger than 93.82% for constant voltage experiments, and larger than and 91.21% constant displacement experiments. High accuracy values are, therefore, obtained in each case.

For comparison purposes, the σ₁ vs. λ₁ curves for different voltages and aspect ratios are reported in figure 17. In here, stress σ₁ is preferred over force f₁ to provide a better comparison with the results in figure 10. The curves only show the quasi-static material response curve, extrapolated by means of the obtained model. As expected, similar results and trends to the simulations study in section 4.1 are obtained. Based on this plot, it can be quantified that the difference in aspect ratio results in stress variations, which range between 15% and 30%, given the same stretching condition.

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As a further validation, a comparative study between the lumped and the FE model is also conducted based on the prediction of the experimentally calibrated Yeoh model. This study will permit to further assess the validity of equation (58) not only for the stress response but also for the kinematic one (i.e. λ₂ and λ₃). Figure 18 portrays the quasi-static σ₁ vs. λ₁ curves (upper row) and λ₂, λ₃ vs. λ₁ curves (lower row) for the considered five aspect ratios, by comparing results of both lumped and FE models. Similarly to section 4.1, also in here the two models show a remarkable agreement. This fact represents a further validation of the barrier function model described by (58). From the stretches plots, it can also be observed that α_a ≤ 0.2 and α_a ≥ 5 represent good threshold values for assuming uniaxial and pure shear assumption to hold true, respectively. This finding is in agreement with the behavior expected from pure shear DE specimens [45].

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An additional validation of the kinematic stretches assumption is conducted by comparing the membrane surface area predicted by the model with the one measurement via the fixed camera on the test bench (as described in section 5.1). To compare experiments and simulations, an area stretch λ_A is introduced, defined as follows:

$$\begin{equation}{\lambda _A} \equiv \frac{a}{A} = {\lambda _1}{\lambda _2}\end{equation} \tag{ 60 }$$

where A and a denote the DE surface area in undeformed and deformed states, respectively. Experimentally recorded area stretches, as well as simulated ones, are reported in figure 19 for all of the five considered aspect ratios, by assuming v = 0 kV. As it can be noticed, the area stretches predicted by both FE and lumped models match the experimental data with considerable accuracy. For further comparison, an example of video frames for different stretch states, as well as their corresponding area reconstructed via image processing and simulated in COMSOL, is reported in figure 20. A remarkable agreement between measured and simulated shapes can be observed. It can then be concluded that the presented model provides an effective way to quantify the effect of necking occurring during membrane stretching.

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