Modelling of an electroactive polymer actuator

Transcription

1 vailable online at Proceia Engineering 48 (1 ) 1 9 MMaM 1 Moelling of an electroactive polymer actuator Ákos czél a * a zéchenyi István University, Egyetem tér 1,Gy r, H-96, Hungary bstract The aim of this paper is to buil a moel of an electroactive polymer actuator when external electric fiel is applie. Materials whose rheological properties can be varie by electric excitation are calle electroactive materials. The behavior of these rubberlike meia is commonly investigate in two ifferent ways. Briganov, Dorfmann, Bustamante, Ogen an others escribe the couple problem of finite eformation of the continua in electromagnetic fiel by taking into account all the electromagnetic phenomena. Pelrine, Kornbluh, ommer-larsen an others present a phenomenological escription. Our aim is to form a brige between these two points of view by neglecting those terms of the governing an transformation equations that are smaller than other contributions by several orers of magnitue. First, those equations are presente that govern the finite eformation an the electromagnetic phenomena insie the material. fter that, those estimations will be taken that show the orer of magnitue of the ifferent contributions to the so-calle effective fiels. Finally, the moel of the EP actuator uner perioically changing electric fiel will be presente. Because of the perioically changing finite eformation of the actuator, the electromagnetic phenomena must be investigate in the rest frames fixe to every single point of the material boy. The electromagnetic fiel-variables can be converte into the laboratory frame by the slow spee approximation of the Lorentz transformation. For the special case of thin electroactive polymer actuators, one can fin that the velocity epenent contributions are smaller by ten orers of magnitue in the electric fiel transformation, an by five orers of magnitue in the magnetic fiel transformation equations. On the other han, none of the terms of the effective current can be neglecte, because they can be of the same orer of magnitue as the free current. 1 The uthors. Publishe by Elsevier Lt. 1 Publishe by Elsevier Lt.election an/or peer-review uner responsibility of the Branch Office of lovak Metallurgical ociety at Faculty election of an/or Metallurgy peer-review an Faculty uner of Mechanical responsibility Engineering, of the Branch Technical Office University of lovak of Košice Metallurgical Open access ociety uner at CC Faculty BY-NC-ND of Metallurgy license. an Faculty of Mechanical Engineering, Technical University of Košice. Keywors: electroactive polymer actuator, effective fiel, Lorentz transformation, constitutive equation Nomenclature c spee of light in the vacuum (m/s) thickness of the EP layer (m) f frequency (1/s) q charge per unit volume (C/m 3 ) area of the EP layer (m ) C capacitance of the EP actuator (F) F force acting on the EP layer ue to the applie voltage (N) I current () U potential energy per unit volume (J/m 3 ) Y Young s moulus (N/m ) V voltage applie to the actuator (V) * Corresponing author. aress: aczela@sze.hu Publishe by Elsevier Lt.election an/or peer-review uner responsibility of the Branch Office of lovak Metallurgical ociety at Faculty of Metallurgy an Faculty of Mechanical Engineering, Technical University of Košice Open access uner CC BY-NC-ND license. oi:1.116/j.proeng

2 Ákos czél / Proceia Engineering 48 ( 1 ) 1 9 r position vector (m) ( n) t stress vector (traction) at the surface with unit normal n (N/m ) v velocity of the material boy (m/s) B magnetic inuction (T) D electric isplacement (C/m ) E electric fiel intensity (V/m) F force per unit volume, acting on the material boy (N/m 3 ) H magnetic fiel intensity (/m) J current per unit area (/m ) L boy couple per unit volume (N/m ) M magnetization ensity (/m) P polarization ensity (C/m ) Q heat flux vector (W/m ) Greek symbols ε ε relative ielectric constant, ielectric constant in the vacuum (1, s/vm), r μr, μ relative magnetic permeability, magnetic permeability in the vacuum (1, Vs/m) ν conuctivity (/Vm) ρ mass ensity (kg/m 3 ) Φ energy supply ensity (W/m 3 ) Vector calculus ab scalar prouct (ot prouct) of two vectors a b vector prouct (cross prouct) of two vectors Nabla, Hamilton s ifferential operator E curle = ε E, i ijk j k curle ( ) ive E, iei V a V volume integral of a over the volume V a surface integral of a over the surface C a C contour integral of a over the curve C a flux of a on the surface 1. Introuction The electroactive materials can eform in response to applie electric fiels. Therefore, the actuators mae from these rubberlike materials have been a promising area of research for more than a ecae now [1-]. Pelrine, Kornbluh, ommer- Larsen an others have presente a phenomenological escription by summarizing an interpreting countless results in measuring [3]. Unfortunately, there are several ifficulties when moeling these rubberlike materials uner finite eformations while excite by electric fiel. Briganov, Dorfmann, Bustamante, Ogen an others escribe this couple problem of finite eformation of the continua in electromagnetic fiel by taking into account all the electromagnetic phenomena [4-6]. In general, it is the perfect metho for escribing all the electromagnetic phenomena. But in some cases, especially, when no external magnetic fiel emerges, an the material boy uner investigation is flat, some effects can be isregare.. The balance equations The mechanical behavior of any material boy is governe by the balance equations of the continuum mechanics [7]. These four basic balance laws are: Balance of mass:

3 Ákos czél / Proceia Engineering 48 (1 ) ρ V = t (1) V Balance of linear momentum: t v V t F V () ρ = ( ) + V V Balance of angular momentum: t r v V = r t + r F+ L V (3) ( n) ρ V V Balance of energy: t 1 ( ) ρ + = ( ) + ( +Φ Q +Φ ) n v v t v Q n F v V V U V V (4) The question is what functions appear as boy force per unit volume F, boy couple per unit volume L, an energy supply ensity Φ of electromagnetic origin. Naturally, all these three functions can be erive from the electromagnetic fiel variables. Unfortunately, the boy force, the boy couple an the energy supply can be variely postulate accoring to the formulation use to escribe the electromagnetic phenomena in the moving meia [8-9]. 3. The Maxwell-equations for meia in rest In vacuum, the physical laws for the electromagnetic fiel-variables are the well-known Maxwell-equations [1]: These local equations can be erive from the global Maxwell-equations: mpère Maxwell D curl H = J + (5) t B curl E = (6) t iv B = (7) iv D = q (8) H C= J + t D (9) C Faraay E C= t B (1) C Gauss Faraay

4 4 Ákos czél / Proceia Engineering 48 ( 1 ) 1 9 Gauss Coulomb B = (11) D = qv (1) V ince these integrals can be measure, only the global laws can be proven by experiments. In the vacuum D= ε E an B = μ H, so two fiel variables are sufficient for escribing all electromagnetic phenomena, one for electric an one for magnetic fiels. The sources of the electric an magnetic fiels are electric charge an electric current, respectively. For material boies, the relationships between the fiel variables (an the current) can be very complex: (, ); (, ); (, ) D= E H B = H E J = E H (13) where ; ; are general vector functions. These equations are often calle as the constitutive equations for the material uner iscussion. For linear isotropic materials the constitutive equations are: D= εe; B = μh; J = ν E, (14) where ε is calle the ielectric constant, μ the magnetic permeability, an ν the electric conuctivity. There is a tenency to regar E an B fiels as the basic variables for electric an magnetic fiels in the vacuum so the constitutive laws can be rewritten by introucing two more variables for material meia: ( ) D= ε E+ P; B = μ H+ M, (15) where P is the polarization ensity an M is the magnetization ensity. Their efinitions are the above two equations. 4. The equations for moving meia In the case of moving meia, the first challenge is to fin the aequate reference co-orinate system. The laboratory frame is regare as inertial system an the intensity of the external electromagnetic fiel is usually given in this representation, so it seems obvious to escribe all the electromagnetic phenomena in the laboratory frame. Unfortunately, the constitutive equations are known only for the material lying at rest. In aition, all the points of the eforming meia may move by the velocity of their own, so there is no co-orinate system in which the material boy uner investigation seems to be at rest. Naturally, one can choose a co-orinate system in which one single point an its small surrounings are at rest an this system may be calle rest frame (Fig. 1.). However, other areas of the boy uner investigation may not be at rest accoring to this so-calle rest frame. Laboratory frame z E H z Rest frame v x y y v Moving meia x Fig. 1. Laboratory frame an rest frame for escribing the moving meia

5 Ákos czél / Proceia Engineering 48 (1 ) The first step of the usually chosen proceure is to write an solve the Maxwell-equations in all the co-orinate systems fixe to every single point of the moving an eforming meia. fter that, the results must be converte into the laboratory frame. The chilles heel of this metho is the transformation from one co-orinate system to an other. Coorinates of the electromagnetic variables must be transforme from one system to the other accoring to the Lorentz transformation. s the velocity is much smaller than the spee of light, the slow spee approximation of the Lorentz transformation can be use: E = E v B D D v H + ; = + c (16) H H v D B B v E = ; = c (17) J = J qv q = q vj c ; (18) The use of these transformation equations is self-evient in vacuum, but there are ifferent commonly use formulations for transformation in the presence of matter. ome of these formulations are iscusse in [8]. It is important to note that they iffer from each other not only in some notations but also in regaring the polarization an the magnetization relevant or irrelevant when efining the effective fiels. The use of Lorentz transformation is problematic in other respect as well: The points of the eforming material boy are moving with not a constant velocity, an no co-orinate system fixe to an accelerating point can be rigorously consiere as inertial system. On the other han, the acceleration can be manage by postulating inertial forces an the relativistic effects can surely be neglecte since the velocities are very small, compare to the spee of light. Further problems emerge when taking into account the electromagnetic effects of the moving charge particles of the eforming material itself [11-1]. Fortunately, in the case of electroactive polymer (EP) actuators no charge particles an no current occur insie the material. 5. The moel of the EP actuator The geometry of the common EP actuators is planar or can be consiere planar (the thickness is much smaller than the curvature of the evice). The mechanical moel of an average EP actuator (Fig..) can be imagine as a flat capacitor: a thin electroactive polymer layer coate on both sie with compliant conuctive film (these are the plates of the capacitor an are sai to be compliant because they can strain together with the polymer even uner finite eformation). The polymer is rubberlike an quasi-incompressible with a Young s moulus of some MPa. ctuation is cause by electrostatic force between the two electroes (Maxwell pressure). This force squeezes the polymer an because of its quasi-incompressibility, it expans in area ue to the electric fiel. Therefore, charge an current can occur only on the surface, or more precisely in the plates of the capacitor. V t = V sin π f t () ( ) complient conuctive films EP layer = 3 m Fig.. Moel of the EP actuator The EPs are excellent insulators so no free charge an no free current can occur insie the material, except for the case of amage [3] There is a lack of magnetisable particles in the EPs, too. These facts simplify the Maxwell equations:

6 6 Ákos czél / Proceia Engineering 48 ( 1 ) 1 9 curl ε E P H = + (19) t t H curl E = () t iv H = (1) iv P = ε ive () The polarization is observe to be the function of the electric fiel intensity, the temperature an the stretch [13]. Hysteresis may also occur so the constitutive equation is eviently nonlinear [14]. In aition, we can assume the lack of external magnetic fiel as the EP actuators are usually use without external magnetic excitation. Unfortunately, it oes not mean that no magnetic phenomena woul take place at all. When an EP actuator is in steay state (that is to say there is no more change in its imensions an electric fiel intensity insie), it is self evient that no current is flowing, so no magnetic fiel emerges. However, before reaching the steay state the shape of the actuator changes in time, so the charge on the plates moves together with the EP s surface. It means current from the point of view of those who are in the laboratory frame, this current results in magnetic fiel, accoring to the mpère Maxwell law. lthough the current is outsie the material boy uner survey, the magnetic fiel cause by this current will affect the whole EP actuator. Furthermore, we have to face the problem of the ifferent co-orinate systems because of the movement of the meia. 6. Electromagnetic an mechanical conitions in the EP actuators The EP film thickness can vary from several m up to some ten m, epening on the original thickness of the film an the prestretch. The imum electric fiel strength can be more than 1 8 V/m [13]. It results in a breakown voltage of some kv. The capacitance an the capacitive energy can be calculate by the well-known formulae: 1 1 C = εε ; EC = V C = V εε, (3) where ε r relative ielectric constant can vary between 3 an 5 epening on the material, the electric fiel intensity an the stretch. The force, what an actuator of this type can exert is: EC V εε F = = (4) The incompressibility of the material has been taken into account ue to the =const. conition. This force is perpenicular to the capacitor s plate an results in compressing the polymer in thickness. The external electric fiel is sinusoial. The higher the frequency is, the more significant magnetic phenomena are expecte. However, there is no mean in increasing it above the usual operating frequency. ccoring to the experiments [3] (page 118), some khz is available as a imum operating frequency. ssuming that the voltage applie on the evice is sinusoial an the capacitance oes not change in time, the current is sinusoial also. This current causes a magnetic fiel insie the polymer. ( π ) It () = π fcvcos ft (5) ( π ft) I π fcvcos H = = ; B = μh (6)

7 Ákos czél / Proceia Engineering 48 (1 ) J B B J V() t Fig. 3. Magnetic inuction ue to the current flowing into the plate of the capacitor For the sake of simplicity, the shape of the capacitor is assume as square. The fiel intensity is parallel with the conuctive layers an perpenicular to the irection of the current (Fig. 3.). In aition, a magnetic fiel is inuce ue to the altering electric isplacement. It equals: D t π fε ε V ( π ) 4 4 H = = cos f t ; B = μh (7) This contribution to the magnetic fiel is also parallel with the layers an reaches its highest amount near the eges of the EP layer (Figure 4.). D V() t Fig. 4. Magnetic inuction ue to the change of the electric isplacement B The above calculations were all mae uner the assumption that there are no movements in the system. The EP is consiere incompressible, therefore the change of area ue to the applie voltage is: F Δ = (8) Y Changing in area goes han in han with the acceleration an velocity of every single point in the material boy. The imum spee occurs at the eges of the actuator: Δ F V εε 4 v = π f = π f = π f (9) 4 Y Y It can increase up to some m/s [9]. With these results, one can estimate the ratios of the terms in the Lorentz transformation: μ Vεεrπ f = v B E Y (3) v H c D Y c εε Vπ f = (31) v D H = V εε Y (3)

8 8 Ákos czél / Proceia Engineering 48 ( 1 ) 1 9 v E V = c B Yc μ (33) ubstituting the abovementione imum operational frequency an breakown voltage, the first two ratios happen to be of the orer of 1-1, the thir an fourth ones of the orer of 1-5. fter these calculations, we can get back to the main question: what functions appear as boy force per unit volume F, boy couple per unit volume L, an energy supply ensity Φ of electromagnetic origin in the balance equations of the continuum mechanics. Boy couple equals zero, as no magnetisable parts can be foun in the material. The abovementione formulations agree in the expression calle Maxwell-Lorentz force an the energy supply expression [6]: F = E+ J B (34) q t t Φ= Jt E (35) qt = q P (36) The only ifference between the formulations is that the ( P ) P Jt = J+ + ( P v) + M (37) t v term is missing in the Maxwell formulation. The polarization fiel is homogenous insie the EP, therefore its ivergence equals zero. In aition, no magnetization takes place at all. Only two terms remaine in the expression of the total current ensity. P t V = ( εr 1) ε π f (38) ubstituting the breakown voltage an the imum frequency, an then multiplying by the area of the actuator, one can get a current comparable with the free current flowing into the electroes. Polarization is assume homogenous insie the material, but velocity is linearly changing from one ege of the actuator to the opposite. Taking notice of only linear islocation: 3 Pv V curl ( P v ) = = ( ε 1) r εεr π f (39) 3 Y It happens to be some percent of the free current. This term emerges only in the Lorentz formulation so this fact can be a basis for eciing by experiments which formulation escribes the real electromagnetic phenomena in the moving meia. Finally, the current ue to the movement of the charge electroes can be estimate as: 3 3 V I = CVv = εε πf 3 (4) Y It is smaller than the free current by five orers of magnitue. 7. Conclusions Before eciing which term to neglect, one has to take into account that the experimental ata of the constitutive equations show high variation. The material properties of the EPs change with frequency, temperature, stretch, an even in time. In these circumstances, it is useless to take into account negligible effects. The calculations of the last section showe us which terms coul be neglecte in the (16-18) formulae of the Lorentz transformation an in the (36-37) formulae of the total charge an total current ensity. ccoring to (3-33), the v B an the v H c terms can surely be isregare. lthough v D an v E c play a role higher by five orers of magnitue, they can be neglecte too. ll results epen

9 Ákos czél / Proceia Engineering 48 (1 ) on the working frequency, the relative ielectric constant, the imum voltage an the thickness of the EP layer. These can vary ue to technological evelopment; therefore, the estimations must be repeate if better materials are achieve. The current ue to the movement of the charge electroes is negligible compare to the free current measure in the rest frame (4). Therefore, it oes not have to be taken into account at all. The polarisation current an the free current are of the same orer of magnitue, so none of them can be neglecte. The curl P v term coul be estimate as some percent of the polarisation current, so it cannot be isregare either. ( ) cknowlegements The research was supporte by the Project BRO-ND7-ND-INRG References [1] Carlson, J. D., Jolly, M. R.,. MR Flui, Foam an Elastomer Devices, Mechatronics, 1, pp [] Koronsky, W., 1993., Magnetorheological Effects as a Base of New Devices an Technologies, Journal of Magnetism an Magnetic Materials, 1, pp [3] Carpi, F. et al., 8. Dielectric Elastomers as Electromechanical Transucers, Elsevier [4] Briganov, I.., Dorfmann,., 3. Mathematical Moelling of Magneto-sensitive Elastomers, International Journal of olis an tructures, 4, pp [5] Bustamante, R., Dorfmann,., Ogen, R. W., 9. On Electric Boy Forces an Maxwell tresses in Nonlinearly Electroelastic olis, International Journal of Engineering cience, 47, pp [6] Dorfmann,., Ogen, R. W., 4. Nonlinear Magnetoelastic Deformations of Elastomers, cta Mechanica, 167, pp [7] Maugin, G.., 1988., Continuum Mechanics of Electromagnetic olis, North-Hollan, msteram, pp [8] Pao, Y. H., 1978., Electromagnetic Forces in Deformable Continua, in: Mechanics Toay, 4., Nemat-Nasser,., Eitor. Pergamon Press, pp [9] czél, Á., 1. Electromagnetic Forces in Electroactive Polymers, cta Technica Jaurinensis 5/1, pp [1] Kuczmann, M., Iványi,., 8., The Finite Element Metho in Magnetics, kaémiai Kiaó, Buapest [11] Jolly, M. R., Carlson, J. D., Munoz, B. C., 1996., Moel of the Behaviour of Magnetorheological Materials, mart Materials an tructures, 5, pp [1] Brashaw, D. H et al., 1. Electromagnetic Momenta an Forces in Dispersive Dielectric Meia, Optics Communications 83, pp [13] Kofo, G. et al., 3. ctuation response of polyacrylate ielectric elastomers, Journal of Intelligent Material ystems an tructures 14/1, pp [14] Vu, D. K., teinmann, P., 7., Nonlinear Electro- an Magneto-elastics: Material an spatial settings, International Journal of olis an tructures, 44, pp