Abstract

A Dielectric Elastomer Generator (DEG) is an electromechanical transducer, basically a highly deformable parallel-plate capacitor, made up of a soft DE membrane coated with two compliant electrodes on its opposite surfaces. This device is able to convert mechanical work, emanating from its interaction with the environment, into electrical energy. The capacitance depends on the deformation undergone by the membrane, and its variability can be exploited to extract electric energy by (i) initially stretching, (ii) then charging the capacitor, (iii) subsequently releasing the stretch and finally (iv) harvesting the charge at a higher electric potential. The optimisation procedure for a load-driven soft planar DEG is presented, assuming hyperelastic and ideal dielectric behaviour. The DEG undergoes the ideal four-stroke electromechanical cycle previously described and its performance is evaluated on the basis of the energy extracted during a cycle and of the efficiency, defined as the ratio of the harvested energy on the total invested energy. The amount of extracted energy is limited due to possible failures of the device, which are, in the most general case, electric breakdown, material rupture, buckling-like instabilities due to loss of the tensile stress state and electromechanical instability. These failure mechanisms determine the allowable state region for the generator. Hence, in order to identify the best cycle that complies with these limits, a constrained optimisation problem is formulated and the generator performance is estimated. For the different loading cases examined, namely equibiaxial stress state and plane strain, numerical results show, as expected, a critical dependence of the harvested energy on the ultimate stretch ratio and, against expectations, a universal limit on the dielectric strength of the DE membrane beyond which the optimal cycle is independent of this parameter. Thus, there is an upper bound on the harvested energy, which depends only on the ultimate stretch ratio. In addition to the simple parallel-plate configuration, an annular DEG deforming out-of-plane has been analysed. In this configuration the generator is made up of an annular membrane constrained at the boundary by a rigid ring and at the centre by a rigid plate, on which an external force is applied. Due to the loading, the membrane deforms non-homogeneously out-of-plane. In order to avoid loss of the tensile stress state, electric breakdown and electromechanical instability, the applied voltage is controlled, thereby limiting the maximum voltage and keeping the maximum stretch in an admissible range. Numerical results show that the prestretch of the membrane is crucial for an effective behaviour of the device. In fact, the unprestretched generator performs poorly with regard to both energy and efficiency. A small prestretch, of approximately 5%, ensure a sixfold improvement in the gained energy and a fivefold increment in efficiency. The performance of the generator is evaluated for different values of the applied load and of the prestretch. This analysis shows that increasing the applied force the harvested energy increases monotonically, while the efficiency increases until a peak value and then decreases. Hence, for an out-of-plane DEG, the choice of the applied force is decisive to ensure a good trade-off among energy and efficiency. Moreover, a comparison of different DEG layouts demonstrates that the annular DEG can compete with the equibiaxial planar generator, in terms not only of efficiency, but also of harvested energy. What has been so far pointed out is valid under the hypothesis of ideal, lossless material. Since polymers are affected by time-dependent effects, this hypothesis appears to be not completely realistic. Indeed, a predicting model for soft dielectric elastomer generators must include a realistic model of the electro-mechanical behaviour of the elastomer filling, the variable capacitor and of the electrical circuit connecting all the device components. To this end, the ideality assumption of the material and of the cycle has to be removed. Hence, a complete framework for a reliable simulation of soft energy harvesters is proposed for a soft viscous dielectric elastomer generator, operating in an electrical circuit for energy harvesting and subjected to a periodic mechanical stretch. The electrical model of the generator takes into account the effects of the electrodes and of the conductivity current through the dielectric material. A phenomenological electro-viscoelastic model at large strain is proposed and calibrated on the basis of experimental data available in literature for a polyacrylate elastomer (VHB-4910). The effects of viscoelasticity and of possible changes of the permittivity with strains on the generator performance are hence investigated. Numerical results underline the importance of time-dependent effects on the evaluation of the generator performance. The main outcome of this analysis is that, compared with a hyperelastic model, the efficiency is reduced by viscoelasticity for high values of the mean stretch and of the amplitude of stretch oscillation. The reduction is almost insensitive of the mechanical frequency while the efficiency is further reduced by the variation of the permittivity with strain. Moreover, viscoelastic effects modify the allowable state region of the generator. At regime condition, the failure curves relative to electromechanical instability and to loss of the tensile stress state are strongly modified by the viscous effects. This fact results in the alteration of the allowable state region of the generator. Furthermore, due to the change in shape and size of the admissible region under this condition, a more surprising result is the fact that the natural configuration is a not-allowed state. As a consequence, there is an upper bound on the maximal stretch oscillation amplitude. Focusing on the main features of the electrical circuit, an important outcome of the analysis is the identification of a value range of the external electric load for which the efficiency is maximal. Furthermore, the viscous dissipation of the material dominates the energy loss arising from the leakage current across the dielectric membrane.