EuroEAP 2014 International conference on Electromechanically Active Polymer (EAP) transducers & artificial muscles
Organized and supported by ‘European Scientific Network for Artificial Muscles – ESNAM’ (www.esnam.eu) COST Action MP1003
Linköping, 10-11 June 2014
Electrostatic-elastodynamic finite element modelling of stacked dielectric actuators
Poster ID:
1.2.15
Tristan Schlögl, Sigrid Leyendecker
Contact e-mail:
[email protected]
University of Erlangen-Nuremberg, Chair of Applied Dynamics, Erlangen, Germany
Abstract Generally, the three-dimensional behaviour of dielectric elastomer actuators (DEA) is covered by the Maxwell equations and the balance of momentum as shown by Dorfmann [1]. Building on a variational finite element formulation of the static coupled problem introduced by Vu [5], in this work inertia terms are added in order to obtain a description of the deformation process depending on time. This allows for a structure preserving time integration of fully three-dimensional DEAs [2]. The obtained scheme is used to simulate stacked actuators at finite strains. A reduction method is introduced that allows for the simulation of thin single layers at low computational cost.
Bionicum project – artificial muscles This joint project addresses on the one hand the development of automated production of multilayer DEAs together with the lightweight power electronics (Institute for Factory Automation and Production Systems, Erlangen Uni), and on the other hand the derivation of a simulation framework to characterise and predict the deformation process and effective forces when voltage is applied to the DEA (Chair of Applied Dynamics, Erlangen Uni) [3]. To gain DEAs with driving voltages below one kilovolt, a reduction of the layer thickness of the dielectric medium below 100 micrometer must be realised. Within the presented project, the Aerosol Jet Printing process is investigated for producing DEA layers with thicknesses ranging from four to twelve micrometer [4]. Furthermore, a lightweight control hardware is developed (see poster 2.2.5) that consists of a central power supply and a logic part, which is able to generate the driving signal for several actuators using pulse width modulation.
𝑬 electric field 𝑫 electric displacement 𝑷 Piola-Kirchhoff stress 𝑷ele electrical stress 𝜌0 density
Electrostatic-elastodynamic coupling Maxwell’s equations (electrostatics) rot 𝑬 = 𝟎 and
div 𝑫 = 0
Stacked actuator
coupled problem
momentum balance (elastodynamics)
div 𝑫 = 0 div 𝑷 + 𝑷ele = 𝜌0 𝒙
div 𝑷 + 𝒃0 = 𝜌0 𝒙 electromechanical coupling 𝒃0 = div(𝑷ele )
Variational principle
Discretisation
discretisation of action instead of weak form → structure preserving integration scheme 𝛿𝑆𝑑 = 0
kinetic energy 1 𝑇= 2
𝜌0 𝒙
2
𝑑𝑉
temporal • midpoint quadrature • finite differences
ℬ0
potential energy Π=
𝜙 electric potential 𝑄 surface charge
Ω 𝑭, 𝑬 𝑑𝑉 + ℬ0
Material parameters
𝜙𝑄 𝑑𝐴 𝜕ℬ0
action integral 𝑡𝑁
𝑆=
𝐿 𝑑𝑡 ,
shear modulus Lamé constant pure electric
𝜇 = 0.233 MPa 𝜆 = 999.8 MPa 𝑐1 = 5 ⋅ 10−8 N V 2
(𝐸 = 0.7 MPa) (𝜅 = 1000 MPa) (𝜀𝑟 ≈ 3)
coupling
𝑐2 = 1 ⋅ 10−9 N V 2
(100 V ≈ 20 % contraction)
𝐿 =𝑇−Π
𝑡0
spatial • finite element hexahedrons • tri-linear shape functions
weak form (Hamilton’s principle) 𝛿𝑆 = 0
Mesh
Material [5] 𝜇 𝜆 Ω 𝑭, 𝑬 = 𝑪: 𝟏 − 3 − 𝜇 ln 𝐽 + ln 𝐽 2 2 𝑫 = −𝜕𝑬 Ω and 𝑷 + 𝑷ele = 𝜕𝑭 Ω
2
1 + 𝑐1 𝑬 ⋅ 𝑬 + 𝑐2 𝑪: 𝑬 ⊗ 𝑬 − 𝜀0 𝐽𝑪−1 : 𝑬 ⊗ 𝑬 2
𝑭 deformation gradient 𝑪 Cauchy-Green strain 𝐽 change in volume
single layer • 150 nodes • 60 elements • 4 ⋅ 150 = 600 degrees of freedom
Outlook
Model reduction reduction leads to significant computational cost savings and only small error
full model, 6 stacked cells, 13 mesh layers 60 ⋅ 13 = 780 elements 0 V .. 100 V
- 99 %
60 ⋅ 6 = 360 elements reduced model 60 elements
0 V .. 100 V 0 V .. 600 V
•
damping via hyperviscoelastic material model
•
DEA as actuator in humanoid system
•
optimal control for DEA driven system
layer replacement active material volume ↑ contraction ↑ electric field strength ↓ contraction ↓
layer reduction, same results, because 𝐄 = 𝐄(𝜕𝐗 ϕ)
Δmax 2 %
References 1. 2. 3. 4. 5.
A. Dorfmann, R. W. Ogden, Nonlinear electroelasticity, Acta Mechanica, Vol. 174, pp. 167-183, 2005. J. E. Marsden, M. West, Discrete mechanics and variational integrators, Acta Numerica, pp. 357- 514, 2001. S. Reitelshöfer, M. Landgraf, T. Schlögl, J. Franke, S. Leyendecker, Qualifying dielectric elastomer actuators for usage in complex and compliant robot kinematics, EuroEAP, Dübendorf, Switzerland, 25-26 June, 2013. T. Schlögl, S. Leyendecker, S. Reitelshöfer, M. Landgraf, I. S. Yoo, J. Franke, On modelling and simulation of dielectric elastomer actuators via electrostatic-elastodynamic coupling, IMSD, Busan, Korea June 30 - July 3, 2014. D. K. Vu, P. Steinmann, G. Possart, Numerical modelling of non-linear electroelasticity, International Journal for Numerical Methods in Engineering, Vol. 70, pp. 685-704, 2006.
