Polymers which respond to electrical input by changing their mechanical properties are often denoted electroactive. Our research group focus on electroactive polymers (EAP) which exhibit electrostrictive response to electric input.

There are many examples of interactions between electromagnetic fields and deformable bodies where electromagnetic input results in mechanical output. Perhaps most well known is the linear piezoelectric effect, which was experimentally confirmed in the late 19th century and was used in technical applications as early as in WWI.

While piezoelectric materials have proven useful for a wide range of technical solutions, the strain response due to the electric input is in general very small. Certain electroactive polymers (EAP) on the other hand have shown great promise for producing giant electric-field-induced deformations, with strain responses in certain experimental setups reaching several hundred percent. In the last decades, interest in these materials has been steadily increasing.

In our research group we focus on electric EAP (with the other main class being ionic EAP), and specifically EAP with a mechanical response which is proportional to the square of the applied electric field strength or electric polarization. A very common application for electric EAP is dielectric elastomer actuators, where the setup is based on thin elastomer films sandwiched between highly compliant electrodes. Large deformations can be obtained by exploiting the Coulomb forces between the charged electrodes, squeezing the elastomer sample between them so that it contracts in thickness and expands in width. For certain materials, inherent electrostriction has also been reported where the strain response is due to other effects than the Coulomb forces.

Applications for EAP include artificial muscles, actuators, sensors, biomimetics, microrobotics, energy harvesting, interactive Braille displays and haptic feedback. The combination of direct conversion between electrical input and mechanical output and other attractive properties of elastomer materials such as light weight, low cost of production and resistance to hostile environments make EAP interesting and suitable for technological applications where more traditional materials and structures may not be possible to use.

The aim of our current research on EAP is to model their electromechanical behavior within a continuum mechanics framework. In solid mechanics, the equations which govern a given problem are the mechanical balance laws together with constitutive relations describing the material behavior. For electromechanically coupled problems, it is also necessary to take the governing equations for the electromagnetic fields into consideration. These are given by the Maxwell equations together with constitutive equations relating the electromagnetic quantities to each other.

Furthermore, the presence of electromagnetic fields in polarizable media alter the mechanical balance laws as they give rise to an additional body force and couple, resulting in lost symmetry of the Cauchy stress tensor. In our research we restrict our considerations to the case of electrostatics, hence the electromagnetic quantities of interest are the electric field, the electric displacement and the electric polarization. The electric field can be thought of as the force exerted on a test charge placed in the field, and has the unit N/C. Similarly, the electric field gives rise to a torque on a dipole or dipole density, as the forces acting on the dipoles negative and positive ends respectively have opposite directions. The resulting electric body force can be considered as the divergence of a stress tensor of electromagnetic origin. When this is added to the Cauchy stress, the resulting total stress tensor is symmetric.

The equations needed to model electromechanical behavior of electric EAP, considering electrostatic conditions, are then given by the balance laws of mechanics, modified to include the electric body force and couple, together with the macroscopic Gauss' law for the electric displacement. It is also necessary to specify constitutive relations for the stress and electric displacement, with the possibility to add electromechanical coupling also at the constitutive level through mixed invariants of the deformation and the electric field.

Restricting the field of study to electrostatics has the major advantage that the electric field in this case can be derived as the gradient of a scalar potential. This makes it possible to use standard nodal-based finite elements for the numerical treatment. The electric potential is simply added as an additional unknown nodal variable in the finite element discretization and the linearized system of equations can be solved in a coupled fashion for displacements and electric potentials.

For more accurate predictions of the electromechanical behavior of EAP applications, it is necessary to take all aspects of the mechanical behavior of the elastomer substrates into consideration. One important such aspect is dissipative effects. For elastomers this is often viscoelastic behavior such as strain rate dependence, stress relaxation and creep.

The viscoelastic behavior of an engineering material affects for instance response time and long-time behavior of its applications. Since EAP are used in technological structures such as actuators and sensors, it may be of great importance to capture any time-dependent effects in a mathematical model. This is illustrated in the figure to the right above. A sample of electrostrictive polyurethane is exposed to a constant voltage for a certain period of time, and the electric loading is then entirely switched off. The experimentally observed values are marked by stars, and it is clear that there is a time dependence in the strain response.

A phenomenological model for the electromechanically coupled behavior of electrostrictive polyurethane was developed by our research group. The model included quadratic coupling between electric field and deformation as well as viscoelasticity in the mechanical part. The coupled behavior and the time-dependent behavior respectively were calibrated using separate experimental data, and then used to predict the time-dependent response as seen in the figure above.

Furthermore, the electro-viscoelastic constitutive model was used in a numerical scheme, where the finite element method was used to simulate some representative boundary value problems. The figures to the left depict the result of simulations of a beam-like structure, where electric potential loading was first applied in a step-wise fashion and then held constant, allowing for the equlibrium state to be recovered. As can be seen in the figures, a large part of the deformation takes place during the constant load phase.

In general in solid mechanics we are faced with the problem of calculating the deformed state of a body exposed to certain loads and boundary conditions, knowing only the shape of the body at rest. In certain cases it may be of interest though to specify the deformed shape of the body beforehand, for instance in applications requiring high precision.

The inverse motion method provides a way to do this for elastic problems. Instead of solving the usual direct motion problem, through a reparameterization of the equations in terms of the inverse deformation it is possible to set up a numerical scheme which takes the deformed, loaded state as input and calculates the corresponding undeformed shape of the body. The numerical framework is a standard finite-element-based method which is essentially of the same format as an updated-Lagrange scheme.

In our reseach, we have extended the original numerical scheme proposed for elastic and hyperelastic problems by Govindjee and Mihalic [Govindjee, S. and Mihalic, P. 1996, 1998] to electroelasticity. Using the phenomenological constitutive model for polyurethane developed previously, but discarding dissipative effects, and the numerical scheme based on the finite element method, a number of representative boundary value problems have been solved demonstrating the efficiency of the formulation and possible uses when developing real-life applications.

The figure below shows the result of inverse-motion-based form finding for a circular electroelastic gripper. The desired deformed state for a given value of the electric potential load is shown to the left. To the right the figures show the calculated undeformed, unloaded configurations for the case without (upper figure) and with additional traction loading (lower figure) on the inner boundary respectively.

[1.] Phenomenological modeling of viscous electrostrictive polymers, Anna Ask, Andreas Menzel and Matti Ristinmaa, International Journal of Non-Linear Mechanics, 47, 156-165, 2012

[2.] Electrostriction in electro-viscoelastic polymers, Anna Ask, Andreas Menzel and Matti Ristinmaa, Mechanics of Materials, 50, 9-21, 2012

[3.] On the modeling of electro-viscoelastic response of electrostrictive polyurethane elastomers, Anna Ask, Andreas Menzel and Matti Ristinmaa, 9th World Congress on Computational Mechanics/4th Asian Pacific Congress on Computational Mechanics, Jul 19-23, 2010, IOP Publishing Ltd

[4.] Inverse-motion-based form finding for quasi-incompressible finite electroelasticity, Anna Ask, Ralf Denzer, Andreas Menzel, Matti Ristinmaa, International Journal for Numerical Methods in Engineering, 94, 554-572, 2013

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