Abstract
The present study investigates nonlinear forced vibration of dielectric elastomerbased microbeam. The nonlinear terms in beam equation are geometric and material one. Geometric nonlinearity is considered by vonKarman strain displacement relationship and the material nonlinearity is modeled with Yeoh hyperelastic model. Galerkin and Multiple scale methods solve the governing equation. This solution that includes primary resonance, leads to frequency response, so that we depict influence of detuning parameter on amplitude in variation of different parameters.
Keywords:
Hyperelastic; Yeoh; Microbeam; EulerBernoulli; Primary resonance
1 INTRODUCTION
Electroactive polymers (EAP) are polymers that undergo size or shape changes when encounter electrical stimulation. Currently these polymers have attracted attention of researchers around the world. Dielectric elastomers (DEs) belong to the group of electroactive polymers. They are lightweight, have a high elastic energy density, and product large strains. Also potential of chemical and biological compatibility, simplicity of structure and robustness due to the use of stable and commercial availability are the other properties that make dielectric elastomers good candidate for applications such as artificial muscles, sensors, generators, loud speakers, micro air vehicles, energy harvesting, haptic surfaces, actuators and resonators (Zhigang, 2010Zhigang, S., (2010). Theory of dielectric elastomers, Acta. Mech. Solida. Sinica. 23: 549578.; Mockenstrum and Goulbourne, 2006Mockenstrum, E.M. and Goulbourne, N., (2006), Dynamic response of dielectric elastomers, Int. Journal Nonlinear Mechanics. 41: 388395.; Feng and Zhang, 2014Feng, C. Yu, L. and Zhang, W., (2014). Dynamic analysis of a dielectric elastomerbased microbeam resonator with large vibration amplitude, Int. Journal Nonlinear Mechanics. 65: 6368.; Carpi et al.,2011Carpi, F., De Rossi, D., Kornbluh, R., Pelrine, R.E., SommerLarsen,P. (2011). Dielectric elastomers as electromechanical transducers: Fundamentals, materials, devices, models and applications of an emerging electroactive polymer technology. Elsevier.; Stoyanov et al., 2009Stoyanov, K., H.G., Gerhard, R., (2009). A CoAxial Dielectric elastomer actuator. Advances in Science and Technology, 61: 8184.; Chakravarty, 2014Chakravarty, U. K., (2014). On the resonance frequencies of a membrane of a dielectric elastomer. Mechanics Research Communications 55.1: 72 76.; Cohen, 2004Cohen, Y.B., (2004). Electroactive polymer (EAP) actuators as artificial muscles: reality, potential and challenges, SPIE Press. ).
DEs are hyperelastic materials and we should consider material nonlinearity in their modeling, The material nonlinearity due to nonlinear stress strain relationship can be modeled using various hyperelastic constitutive models. These models can be classified into two main formulations based on the strain energy functions: (1) phenomenological models such as Mooney, MooneyRivlin, Bidermann, Yeoh, and Ogden model. (2) Physicallybased models such as NeoHookean, Isihara model, and SlipLink model (Markmann and Verron, 2006Markmann, G.,Verron, E., (2006). Comparision of hyperelastic models for rubberlike materials, Rubber Chemistry and Technology. 79.5: 835858.). These models were used by some researchers in different cases. Here the most important works have been reviewed.
(Yeoh, 1990Yeoh, O.H. (1990), Characterization of elastic properties of carbonblackfilled rubber vulcanizates, Rubber Chemistry and Technology, 63: 792805.) represented a strain energy function which was cubic in I _{1}. This function proposed for the characterization of the elastic properties of carbonblackfilled rubber vulcanizates and included variable shear deformation. The use of proposed strainenergy function has been shown to permit the prediction of stressstrain behavior in different deformation modes, from data obtained in one simple deformation mode.
(Ogden and Roxburgh, 1993Ogden, R.W. and Roxburgh, D.G., (1993). The effect of prestress on the vibration and stability of elastic plates. Int. Journal of Engineering Science., 31 pp. 1611 1639. ) studied plane vibrations superimposed on the homogeneous deformation of a rectangular block of incompressible isotropic elastic material. They obtained frequency equations in respect of a general form of strainenergy function. Their interest was the case of zero frequency, because of occurrence of bifurcation equations that each of them determines a set of values of deformation and stress. Also they illustrated dependency of frequency on the deformation, stress and aspect ratio.
(Verron et al., 1999Verron, E., Khayat, R.E., Derdouri, A., Peseux, B., (1999). Dynamic inflation of hyperelastic spherical membrane. Journal of Rheology. 43.5: 10831097.) studied dynamic inflation of hyperelastic spherical membranes with MooneyRivlin material. In addition, they examined the conditions for oscillatory inflation around the static fixed point and found that the frequency of oscillation reaches a maximum at some pressure level, which tends to increase for materials with NeoHookean material.
(Ogden et al., 2004Ogden, R.W., Saccomandi, G., Sgura, I. (2004). Fitting hyperelastic models to experimental data. Computational Mechanics, 34.6: 484502.) concerned determining material parameters in incompressible isotropic elastic strainenergy functions based on a nonlinear least squares optimization method by fitting data from the classical experiments of Treloar and Jones on natural rubber. They considered three separate forms of strainenergy function: principle stretches, principle invariants of CauchyGreen deformation tensor and a certain set of orthogonal invariants of the logarithmic strain tensor. They showed that at least in the case of elastomeric materials, it is advisable to examine their behavior first via mathematical models rather than by simulation models.
(Mason and Maluleke, 2007Mason, D.P., Maluleke, G.H. (2007). Nonlinear radial oscillations of a transversely isotropic hyperelastic incompressible tube. Journal of Applied Mathematics and applications. 333.1: 365380.) showed that when the strain energy function is specified as MooneyRivlin strain energy function, three second order equations are derived for radial oscillations in a thinwalled tube. In addition, they concluded that existence of one Lie point symmetry in presence of time dependent net applied force is only for special forms of the generalized MooneyRivlin strain energy function and for special net applied surface pressure.
(LopezPamies, 2010LopezPamies, O., (2010). A new I1  based hyperelastic model for rubber elastic materials, Comptes Rendus Mechanique, 338:211.) proposed a new hyperelastic model, which was applicable over the whole range of deformations. His model like Yeoh model was based on first invariant strain but with a real number as a power. The proposed storedenergy function in his research constituted a practical platform from which to account for more levels of complexity to model rubber solids.
(Zhu et al., 2010Zhu, J. Cai, S. and Suo, Z,. (2010). Resonant behavior of a membrane of a dielectric elastomer, Int. J. Solids Struct. 47: 32543262.) analyzed nonlinear dynamics of a membrane of a dielectric elastomer. Some of their concluding remarks are as follow: when the fundamental natural frequency vanishes, state of equilibrium would be unstable; the natural frequencies of dielectric elastomers were tunable by varying the prestretch, pressure, or voltage, when driven by a sinusoidal voltage; the membrane resonated at multiple values of the frequency of excitation when driven by a sinusoidal voltage.
(Steinmann et al., 2012Steinmann, P. Mokarram. H, Gunnar,P. (2012), Hyperelastic models for rubberlike materials: consistent tangent operators and suitability for Treloar’s data. Arch Appl Mech, 82: 11831217.) reviewed fourteen selected of hyperelastic models and derived analytical stressstretch relations for certain homogeneous deformation modes. Furthermore they evaluated validity of the models with regard to the classical experimental data on vulcanized rubber published by Treloar. Also they derived ingredient necessary for their fitting procedures for all fourteen models.
(Patil and Dasgupta, 2013Patil. A. . DasGupta, A,. (2013). Finite inflation of an initially stretched hyperelastic circular membrane, European Journal of MechanicsA/solids . 41: 2836) studied finite axisymmetric inflation of an initially stretched flat circular hyperelastic membrane. Membrane’s material was isotropic MooneyRivlin solid. The aim of their work had been to study the inflation of hyperelastic membrane and distinguish between the effects of prestretch and internal pressure on the inflation mechanics. They observed that softening or stiffening of a membrane and the limit point pressure depends on material model and prestretch. Their solution could be used to study inflation assisted thermoforming and angioplasty.
(Breslavsky et al., 2014Breslavsky, I.D. Amabili, M Legrand, M., (2014). Physically and geometrically nonlinear vibrations of thin rectangular plates. Internationa Journal of NonLinear Mechanics, 58: 3040.) investigated static deflection, free and forced vibrations of thin rubber plates. This rectangular plate was under uniformely distributed pressure. Physical and geometrical nonlinearity were introduced by neohookean model and von Karman strains, respectively. Comparison of their result with exact solution, validated corresponding results. They showed that sensitivity of the deflection to the physically induced nonlinearities at moderate strain is significant.
(Feng et al., 2014Feng, C. Yu, L. and Zhang, W., (2014). Dynamic analysis of a dielectric elastomerbased microbeam resonator with large vibration amplitude, Int. Journal Nonlinear Mechanics. 65: 6368.) studied the dynamic properties of a dielectric elastomer (DE)based microbeam resonator with ambient pressure that was affected by using the squeezefilm theory. They approximated analytical solutions for the quality factor and the resonant frequencies. Their results exhibited that the ambient pressure has significant effects on the Qfactor and the resonant frequency shift ratio.
(Bhattacharyya et al., 2015Bhattacharyya, R. Sarangi, S. Samantaray, A.K. (2015), Effect of stresssoftening on the ballooning motion of hyperelastic strings. International Journal of Engineering Scienece. 96: 1933. ) did their research in particular about instability of the steady state balloons. The material was neoHookean rubber. They used shooting method for obtaining steady state results. Theoretical results showed that the single loop balloons have higher tension and they are always stable. Also they depicted the effect of material nonlinearity in steady state curves.
Pineda et al. [20] used from hyperelastic polymer for the improvement of soft sensors with large deformation. They combined hyperelastic materials and conductive fluid to allow the expansion of sensors with enormous deformation. They demonstrated that these sensors are reliable even after several cycles.
(RodriguezMartinez et al., 2015RodriguezMartinez, J.A, FernandezSaez, J, Zaera, R. (2015). The role of constitutive relation in the stability of hyperelastic membranes subjected to dynamic inflation. International Journal of Engineering Science, 93: 3145.) investigated dependency of stability of hyperelastic spherical membrane on constitutive models. They considered six strainenergy functions: three of the MooneyRivlin type and three of the Ogden type. Significant influence of constitutive models on mechanical stability of rubberlike balloons subjected to dynamic loadings was shown. They observed that the steady response of the membrane is extremely conditioned by the strainenergy function selected. In addition they discussed about reduction of uncertainty surrounding the dynamic modeling of rubberlike materials.
(Danaee Barforooshi and Karami Mohammadi, 2016Danaee. Barforooshi, S., Karami Mohammadi, A. (2016). study neoHookean and Yeoh hyperelastic models in dielectric elastomerbased microbeam resonators. 13: 18231837.) considered a microbridge resonator with geometric and material nonlinearity. Geometric nonlinearities were introduced by vonKarman strains and for material nonlinearity the Yeoh and neoHookean models were used. They showed that neoHookean model is not suitable for this case, because of insufficient terms in its strainenergy function. Perturbation technique was used for solution of the nonlinear governing equation and good agreement was achieved between analytical and numerical results. The results showed the significant influence of mode number on the normalized frequencies.
(Madireddy et al., 2015) studied the use of Bayesian approach to estimate the hyperelastic material parameters for sift tissue. They considered MooneyRivlin, exponential and Ogden model to represent hyperelastic nature of soft tissue. They showed that the quality of the fit alone is inadequate to find out the sufficiency of the model. Also, they observed the importance of taking into account the noise and uncertainty in the data in constructing material models.
In this work, dielectric elastomerbased microbeam is considered as a resonator. The Yeoh hyperelastic model is used to describe the material nonlinearity and the vonKarman strains are used to consider large deformations. This microbeam is simply supported at the ends and Multiplescale method is used for solving governing equation.
2 EQUATION OF MOTION
As shown in Fig.1, simply supported Dielectric elastomerbased microbeam has length L, height d, width b.
The equation of motion is derived under the following assumptions: (1) the crosssection of microbeam is constant over the entire length; (2) the extended EulerBernoulli beam theory can be applied; (3) geometric and material nonlinearities are considered due to large deformation and hyperelastic behavior, respectively.
Displacement field for a EulerBernoulli beam is:
where z is the vertical distance of every point of the beam to neutral axis, u is the axial displacement of the point, and w denotes the lateral deflection of the beam.
According to large deformation, GreenLagrange strain deformation is used:
Then, the vonKarman strain components are:
Yeoh developed a hyperelastic material model that only depends on the first strain invariants. This model is based on a series expansion and its series is truncated after the first three terms. Therefore, its strain energy density function is:
Where _{ci} are material constants and I _{1} is the first strain invariant that is related to principle stretches and right Cauchy Green strain tensor as follows:
It should be mentioned that _{λi} (i = 1, 2, 3) are square root of the right Cauchy Green strain tensor (C) and Cis related to strain tensor, E, that its components are introduced in equation (3).
Using potential energy as
Substituting from equations (4), (5) and (6) into (7), leads to the potential energy as a function of w. Also, kinetic energy can be written as follows:
Substituting potential and kinetic energies and considering a distributed frequency force into extended Hamilton principle, leads to following governing equation:
with boundary conditions w(0) = 0,w(L) = 0, δw/δx(0)=0, δw/δx(L)=0.
Equation of motion and boundary conditions can be normalized with proper nondimensional parameters:
ω_{dim} is the linear dimensional natural frequency of microbeam that is achieved by linearization of equation (9).
Therefore, we have:
where
and normalized boundary conditions w(0) = 0,w(1) = 0, δw/δx(0)=0, δw/δx(1)=0.
Reduced Order Model
For solving nonlinear PDE equation, first the Galerkin method is used to reduce the equation to ODE one. So the function will be introduced as multiplication of two separated spatial and time functions as:
Now with multiplying mode shape function in each side of equation (11) and integrating from zero to one, the achieved ODE equation will be:
where
Multiple Scale Method
One of the strong perturbation methods for solving nonlinear equations is the wellknown multiple scale method (Nayfeh, 1979Nayfeh, A.H. (1979). Nonlinear oscillations, John Wiley and sons, New York.). To use this approach, small perturbation parameter, q = εu, is introduced, moreover, the scaled time can be introduced as:
The proposed solution is:
Substituting equations (16) and (17) into equation (14) and arranging based on different orders of _{εi} by Maple software; the following perturbation equations are obtained:
For eliminating small divisor terms, the force term should be placed in the largest order of ε . It should be noted that α^{1} is equal to square of nondimensional linear frequency (α_{1} = ω _{0} ^{2})
The solution of equation (18) can be supposed as:
where cc stands for “complex conjugate ” of the first term.
Substituting equation (21) into equation (19) and eliminating secular terms, leads to:
Now substitution of equations (21) and (23) into equation (20) yields:
To consider primary resonance, we introduce a detuning parameter σ defined by:
And when σ → 0, excitation frequency tends to linear natural frequency. Then
Substituting equation (26) into equation (24) and eliminating secular terms, results in:
Consider the polar form for A:
And substituting equation (28) into (27), multiplying each side by _{eiβ} and finally separating real and imaginary parts, yields:
We transform equations (29) into an autonomous system by introducing the transformation:
And taking derivative yields:
Substituting equations (30) and (31) into equations (29) and after some algebra, the frequency response is obtained as:
It should be noted that in steady state solution: a’=y’ =0
Analytical solution of governing nonlinear ordinary differential equation with substation of equations (21) and (23) into (17), will be:
Mode Shapes and Frequency Response Curves
In this section, a case study has been prepared using geometrical and material properties introduced in table (1). Material constants are based on uniaxial tension test (Martins et al, 2006Martins, P. A. L. S. R, Natal Jorge, M, and Ferreira, A. J. M. A., (2006). Comparative Study of Several Material Models for Prediction of Hyperelastic Properties: Application to SiliconeRubber and Soft Tissues, , Strain Journal, 42: 135147.).
Respect to relation (25) the first three nondimensional natural frequencies, Ω^{*}, at σ = 0 attain 0.0717, 0.1448 and 0.2207, respectively.
Figure (1) shows midpoint displacement versus nondimensional length for different modes. As it is seen, the ends of the microbeam are in agreement with boundary conditions. It should be mentioned that nondimensional maximum amplitude is 0.3 (w ^{*} _{max} =0.3 ) for large deflection of the midpoint and is introduced by initial conditions.
The frequency response curves in nondimensional amplitudedetuning parameter plane are presented around its first natural frequency. These curves depicted by spanning the excitation frequency (Ω) around the first linear natural frequency. With the aid of these curves, the hardening or softening types of behavior in this microbeam can be determined.
Figures (3), (4) and (5) show variation of midpoint amplitude versus detuning parameter for the first, second and third mode of transverse motion, respectively. These figures are depicted based on frequency response introduced in equation (32). As it is seen, as the detuning parameter approaches to zero, the amplitude of motion is increased because of nearness of excitation frequency to the linear natural frequency. This rising continues until the first limit point bifurcation at the top and reaches to maximum amplitude. At this point the system response will be unstable. Stable parts are shown with solid line and the unstable one is shown with dash line. From these figures, it is clear that the system behavior is of hardening type. The maximum amplitude in the figures (3), (4) and (5), at the resonance frequency tends to reach 2205, 777 and 421, respectively.
In addition, the higher mode number is related to hardener behavior of microbeam.
Figures (6)(8) show bifurcation diagrams for different lengths in terms of constant thickness. It is obvious that maximum amplitude is larger and resonance is more intensive in beams with higher aspect ratio (L/d ).
For all modes, the behavior of microbeam gets hardener as the length gets shorter (smaller aspect ratio).
The maximum amplitude at the resonance for the L = 50µm, 40µm and 30 µm in figures (6), (7) and (8) tends to reach [4747, 3396 and 2205],[1676, 1199,777] and [911, 651, 421], respectively.
In figures (9) and (10), influence of the nondimensional amplitude of external harmonic force on the bifurcation diagram is depicted. Two different quantities are chosen here for the force amplitude in each mode. One can observe that for larger force amplitudes, the first stable part is began to rise at smaller detuning parameter and the second stable part is terminated in larger detuning parameter. It means that maximum amplitude occurs at higher excitation frequencies. In addition, the forcing amplitude leads to larger hardening behavior at higher modes.
The maximum amplitude at the resonance for the first and second mode in figure (9) tends to reach 2204 and 777, respectively.
3 CONCLUSIONS
A microbeam with material in the class of Electroactive polymers (Dielectric elastomer) was the case in this study. The nonlinearity that considered here was geometric and material one. Geometric nonlinearity modeled via vonKarman straindisplacement relationship and Material nonlinearity was based on Yeoh hyperelastic model. The nonlinear governing equation solved by Galerkin and Multiplescale methods. Influence of different parameters on amplitudedetuning parameter diagram at primary resonance was studied. Some conclusions and remarks that have been evolved in this article:

Dielectric elastomerbased microbeam with Yeoh model has hardening behavior.

Higher mode number has a direct effect on hardening behavior of DE microbeam.

Higher of mode number has an inverse effect on amplitude of motion.

Higher aspect ratio leads to higher amplitude of motion at all modes.

Shorter beams have more harden behavior.

Magnitude of force amplitude has direct effect on the rise beginning of bifurcation diagram.

The forcing amplitude leads to larger hardening behavior at higher modes.
References:
 Bhattacharyya, R. Sarangi, S. Samantaray, A.K. (2015), Effect of stresssoftening on the ballooning motion of hyperelastic strings. International Journal of Engineering Scienece. 96: 1933.
 Breslavsky, I.D. Amabili, M Legrand, M., (2014). Physically and geometrically nonlinear vibrations of thin rectangular plates. Internationa Journal of NonLinear Mechanics, 58: 3040.
 Carpi, F., De Rossi, D., Kornbluh, R., Pelrine, R.E., SommerLarsen,P. (2011). Dielectric elastomers as electromechanical transducers: Fundamentals, materials, devices, models and applications of an emerging electroactive polymer technology. Elsevier.
 Chakravarty, U. K., (2014). On the resonance frequencies of a membrane of a dielectric elastomer. Mechanics Research Communications 55.1: 72 76.
 Cohen, Y.B., (2004). Electroactive polymer (EAP) actuators as artificial muscles: reality, potential and challenges, SPIE Press.
 Danaee. Barforooshi, S., Karami Mohammadi, A. (2016). study neoHookean and Yeoh hyperelastic models in dielectric elastomerbased microbeam resonators. 13: 18231837.
 Feng, C. Yu, L. and Zhang, W., (2014). Dynamic analysis of a dielectric elastomerbased microbeam resonator with large vibration amplitude, Int. Journal Nonlinear Mechanics. 65: 6368.
 LopezPamies, O., (2010). A new I_{1}  based hyperelastic model for rubber elastic materials, Comptes Rendus Mechanique, 338:211.
 Markmann, G.,Verron, E., (2006). Comparision of hyperelastic models for rubberlike materials, Rubber Chemistry and Technology. 79.5: 835858.
 Martins, P. A. L. S. R, Natal Jorge, M, and Ferreira, A. J. M. A., (2006). Comparative Study of Several Material Models for Prediction of Hyperelastic Properties: Application to SiliconeRubber and Soft Tissues, , Strain Journal, 42: 135147.
 Mason, D.P., Maluleke, G.H. (2007). Nonlinear radial oscillations of a transversely isotropic hyperelastic incompressible tube. Journal of Applied Mathematics and applications. 333.1: 365380.
 Mockenstrum, E.M. and Goulbourne, N., (2006), Dynamic response of dielectric elastomers, Int. Journal Nonlinear Mechanics. 41: 388395.
 Nayfeh, A.H. (1979). Nonlinear oscillations, John Wiley and sons, New York.
 Ogden, R.W. and Roxburgh, D.G., (1993). The effect of prestress on the vibration and stability of elastic plates. Int. Journal of Engineering Science., 31 pp. 1611 1639.
 Ogden, R.W., Saccomandi, G., Sgura, I. (2004). Fitting hyperelastic models to experimental data. Computational Mechanics, 34.6: 484502.
 Patil. A. . DasGupta, A,. (2013). Finite inflation of an initially stretched hyperelastic circular membrane, European Journal of MechanicsA/solids . 41: 2836
 Pineda, F, Bottausci, F, Icard, B, Malaquin, L, Fouillet, Y. (2015). Using electrofluidic devices as hyperelastic strain sensors: Experimental and theoretical analysis. Microelectronic Engineering, 144: 2731.
 Ritto, T.G, Nunes, L.C.S, (2015). Bayesian model selection of hyperelastic models for simple and pure shear at large deformations. Computer and Structures, 156: 101109.
 RodriguezMartinez, J.A, FernandezSaez, J, Zaera, R. (2015). The role of constitutive relation in the stability of hyperelastic membranes subjected to dynamic inflation. International Journal of Engineering Science, 93: 3145.
 Steinmann, P. Mokarram. H, Gunnar,P. (2012), Hyperelastic models for rubberlike materials: consistent tangent operators and suitability for Treloar’s data. Arch Appl Mech, 82: 11831217.
 Stoyanov, K., H.G., Gerhard, R., (2009). A CoAxial Dielectric elastomer actuator. Advances in Science and Technology, 61: 8184.
 Verron, E., Khayat, R.E., Derdouri, A., Peseux, B., (1999). Dynamic inflation of hyperelastic spherical membrane. Journal of Rheology. 43.5: 10831097.
 Yeoh, O.H. (1990), Characterization of elastic properties of carbonblackfilled rubber vulcanizates, Rubber Chemistry and Technology, 63: 792805.
 Zhigang, S., (2010). Theory of dielectric elastomers, Acta. Mech. Solida. Sinica. 23: 549578.
 Zhu, J. Cai, S. and Suo, Z,. (2010). Resonant behavior of a membrane of a dielectric elastomer, Int. J. Solids Struct. 47: 32543262.
Publication Dates

Publication in this collection
Mar 2017
History

Received
27 Aug 2016 
Reviewed
05 Feb 2017 
Accepted
17 Feb 2017