On the Modelling and Computation of Magneto-Sensitive-Elastomers
Final Report Abstract
Magneto-sensitive elastomers are smart materials composed of a rubber-like basis matrix filled with magneto-active particles. Their ability to deform significantly, i.e. geometrically nonlinear, under the stimulation by magnetic fields makes them interesting composites for the development of novel actuators. The principle focus of this research project is the mathematical and numerical treatment of the magneto-mechanical problem in the geometrically nonlinear context. Three physical phenomena pose a challenge: (i) The magnetic field disperses in the material and the surrounding free space. The fields do not cease at the boundary of the body. Therefore, the magnetic field equations have to be considered within matter and outer space. (ii) The normal and tangential components of the magnetic fields fulfill certain jump and continuity conditions along material interfaces. Therefore, appropriate finite element approaches have to be considered to capture this interplay of continuities and discontinuities. (iii) The material exhibits complex constitutive behavior. Therefore, a suitable material model has to be considered to represent the underlying coupling and dissipative effects. In this work, we simulated the geometrical and constitutive non-linear behavior of magneto-sensitive material under the condition of non-mechanical stimuli. We developed a mathematical and numerical model, which captures the coupled nature of the problem. Numerical simulations were performed and their outcome analyzed. The main results obtained were: 1. Variational principles for one-, two-, and three field functionals were derived for magneto-elasticity. The careful definition of the underlying function spaces sets the baseline for the right choice of finite elements approaches in view of a numerical solution. 2. Numerical investigations on a hyperelastic, coupled constitutive model established a guideline for the choice of physically reasonable material parameters. These insights were transferred to the development of a coupled visco-elastic material model. The influence of the magnetic field on the viscous response was highlighted. 3. An algorithm was developed, which allows for a staggered solution scheme, when considering a ponderable body in combination with the surrounding free space. A mesh moving technique was implemented and the achieved results show that the influence of the free space has definitely to be taken into account. It cannot be neglected, especially for materials with relative magnetic constants close to that of vacuum, which is the case for MAPs considered in this work. 4. The numerical simulations were realized with a scalar and a vector potential approach. While the first works within a standard nodal based finite element approach, the latter uses edge based elements for the discretization of the potential. A mixed element was proposed to allow for a monolithic solution of the deformation and the vector potential. The singularities in the magnetostatic system matrix were eliminated with the help of co-tree gauging. The method was validated for linear magnetostatics and applied to a large strain magneto-mechanically coupled problem. The simulation of magneto-sensitive materials, which we presented in this work aims at advancing the understanding for the modeling of smart materials. This work shows that the developed modeling techniques yield promising results. Further research can address some of the limitations and, thus, improve computer-aided design capabilities for engineers in the future. Ultimately, this research field has the potential to accelerate a wider-spread use of smart materials in its broad range of current and future applications.
Publications
- Numerical simulation of particles movement in cellular flows under the influence of magnetic forces. PAMM, 11:573-574, 2011
M. Hriberšek, P. Steinmann, J. Ravnik, F. Vogel
- “On some mixed variational principles in electroelastostatics” Int. J. of Nonlinear Mech. 47.2 (2012), 341-354
Vogel, F., Bustamante, R., and Steinmann, P.
(See online at https://doi.org/10.1016/j.ijnonlinmec.2011.08.001) - “On some mixed variational principles in magnetoelastostatics” Int. J. of Nonlinear Mech. 51 (2013), 157-169
Vogel, F., Bustamante, R., and Steinmann, P.
(See online at https://doi.org/10.1016/j.ijnonlinmec.2012.12.005) - “Modeling and Simulation of Viscous Electro-Active Polymers” Eur. J. of Mech. A. Solid, 48, November–December 2014, Pages 112-128
Vogel, F., Göktepe, S., Steinmann, P., and Kuhl, E.
(See online at https://doi.org/10.1016/j.euromechsol.2014.02.001)
