Issue 23

R. Vertechy et alii, Frattura ed Integrità Strutturale, 23 (2013) 47-56; DOI: 10.3221/IGF-ESIS.23.05 50 where 2 2 2 0 ( ) M m p r r t       is the DE volume, 4.5 8.85 12 / e F m     is the dielectric permittivity of the acrylic film, and 0.6   is a suitable dimensionless correction factor. This expression is based on the assumption that the incompressible DE is a right circular conical horn with constant wall thickness in any of its deformed configurations. As for the DE viscoelastic response, a possible approach is to consider the force response due to a step change in displacement and to superimpose each contribution of a displacement history, ( ) x t , by applying a proper superposition principle. Resorting to a one dimensional model, the overall force response is then given by: ,1 0 0 ( ) ( ) ( ) [ ( ) ( ) ˆ ˆ ] t t ve dx F t t d x t d d                (3) having assumed 0 x  for 0 t  and a differentiable displacement history. The function ˆ ( , ) t x  is named relaxation function and specifies the force response to a unit step change in displacement. In the QLV framework [7,8], the relaxation function takes the form: ,1 ( , ) ( ) ( ) ( ˆ 0) 1 e x t F x g t with g     (4) where F ( e,1) ( x) is the elastic response , i.e. the force generated by an instantaneous displacement, whereas g(t) , called reduced relaxation function , describes the time-dependant behavior of the material. As for the latter term, it is customary to assume a linear combination of exponential functions, the exponents ν i identifying the rate of the relaxation phenomena, and the coefficients c i depending on the material: 0 0 ( ) 1 i r r t i i i i g t c e with c          (5) where, in general, 0 0   . Finally, the total force at the instant t is the sum of the contributions due to all past changes [13], i.e. ,1 ,1 0 [ ( )] ( ) ( ) ( ) t e ve F x x F t g t d x             (6) ,1 0 ( ) [ ( )] ( ) t e g t K x x d         (7) where ,1 ,1 ( ) [ ] / e e K x F x x    . By substituting Eqs. (4) and (5) in Eq. (7), one obtains: ( ) ,1 ,1 0 1 0 ( ) ( ) ( ) i t r t ve e i i F t K x c c e x d                     (8) In particular, referring to Fig. 4, the force response given by the QLV model can be interpreted as that of a nonlinear stiffness connected by a series of r linear Kelvin models (i.e. a parallel spring-damper system). Figure 4 : Actuator non-linear lumped parameter model.