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Authors: Rambausek, Matthias
Title: Magneto-electro-elasticity of soft bodies across scales
Issue Date: 2020
Publisher: Stuttgart : Institute of Applied Mechanics Dissertation xiii, 305
Series/Report no.: Publication series of the Institute of Applied Mechanics (IAM);4
ISBN: 978-3-937399-52-2
Abstract: This thesis is concerned with continuum magneto-electro-elasticity in a quasi-static setting at large deformations. In other words, it is concerned with the description of the mechanical, electric and magnetic response of bodies under mechanical forces, electric field and magnetic field. The interesting part thereby is that magnetic fields cause deformation or changes in the electric polarization of a body, i.e. coupling phenomena. The overall topic, but also the sub-fields electro-elasticity and magneto-elasticity are actively researched in experiment, theory and simulation. In a more general context, applications range from micro-electronic devices such as computer memory, to sensors and actuators over display technology to artificial muscles. The first project embraced by this thesis was the modeling of magnetorheological elastomers (MREs) - a type of composite that combines soft rubber with magnetic particles - across scales. In that course a number of experimentally observed effects, i.e. change of stiffness under magnetic field (Jolly et al., 1996) and deformation with response to magnetic field (Böse et al., 2012) in dependence of microscopic features of the composite (Danas et al., 2012) have been confirmed. Moreover, the conducted two-scale simulations shed light on what is called “shape effect”: for instance, the influence of a specimen’s shape on its the deformation response to magnetic fields. Somewhat counter-intuitively, bodies with given magnetization deform differently for different initial shapes even without mechanical forces being applied. This is not only an interesting phenomenon but a huge challenge for the experimental determination of material properties (Martin et al., 2006; Diguet et al., 2010; Bodelot et al., 2018) which are supposed to be local properties independent of specimen shape. One outcome of this thesis are suggestions on the design of experiments and insight on how to estimate the non-local effect of shape dependence from measurements. Since MREs consist of an elastomer with dielectric properties and inclusions which can be regarded as either dielectric or conducting, it is a natural extension to also include electrostatics. The hope was to observe pronounced magneto-electric coupling, which as opposed to electro-magnetic coupling also exists in quasi-static settings. The most apparent application is the construction of magnetic field sensors (Martin et al., 2003). Interestingly, a phenomenon called magnetoreception - the ability of biological organisms to use (the earth’s) magnetic field for orientation and navigation - is suspected to be rooted in a similar setting (Liu and Sharma, 2013). As reported in the thesis, the above mentioned shape effects play a big role in magneto-electric (ME) coupling. In fact, even materials without classical ME coupling coefficients can be employed for ME devices. This has been demonstrated for ellipsoidal bodies by analytical means as well as by finite element simulations. Another setting investigated are capacitor-like devices where the dielectric medium that resides between the electrodes also features magnetic properties, e.g. a MRE between electrodes. Of course, also for such devices significant ME coupling can be observed. However, there are challenges in the numerical simulations: The geometry of the electrodes locally leads to extreme field concentrations which are difficult to capture by finite elements. Even worse, the unavoidable discretization errors yield electro-mechanical artifacts that potentially affect the accuracy and also the robustness of simulations. One such artifact are “spurious” deformations due to electric or magnetic fields in media which do not feature respective coupling properties. The present thesis does not provide a cure for these issues but at least some benchmark problems which hopefully prove useful for the deeper analysis and eventual resolution of this effect. One attempt to remove or at least reduce the spurious coupling was to go from commonly preferred scalar-potential formulations for electro- and magnetostatics to vector-potential and mixed formulations. Both of them can in fact be regarded as instances of linearly constrained minimization principles which distinguishes them from pure scalar potential formulations. As reported, these alternative formulations indeed seem to increase robustness but fail to provide a true solution. Another outcome of the investigations of alternative finite elements formulations for magneto-electro-elasticity was the observation that mixed formulations are potentially superior over vector potential-based formulations in terms of computation time and memory requirements while their solutions belong to closely related or even equivalent discrete function spaces. The reported observations are based on the use of direct linear solvers for the coupled system of equations. The extension of these studies to the case of iterative linear solvers requires efficient preconditioners of which the construction is not trivial and for now remains a future task. The constrained minimization principles leading to mixed formulations for magneto-electro-elasticity also gave rise to another development. The linear constraints in that formulation are linear partial differential equations in space. Their finite element discretization leads to sparse matrices showing up in the off-diagonal blocks of the Karush-Kuhn-Tucker (KKT) matrix of the magneto-electric-elastic saddle point problem. But all they do is removing the divergence from a vector field. It was recognized that this operation, the projection of a general vector field to the space of divergence free vector fields is a mode-wise local operation in Fourier space and thus much cheaper than in finite elements. The result is the Fourier-based solver for magneto-mechanical problems at microscopic scales for the computation of effective material responses for the case of vanishing free currents. The latter restriction is, however, not an issue in the microscale modeling of MREs, which can be modeled an insulators. A particular appeal of the Fourier-type solver developed is the projection-based approach which has solid mathematical foundations, most importantly the Helmholtz decomposition theorem. Thanks to its structure the method can be directly extended to magneto-electro-elasticity in absence of free charges. A possible question for future investigations is how to also account for free current and charge densities. Besides the studies above the thesis also revisits the theoretical foundations magneto-electro-mechanics. Accompanying the application-oriented work, the theoretical part is quite extensive with the intention to provide a solid basis for future research beyond the quasi-static regime and beyond bulk materials in Euclidean space, e.g. films and shell structures or “partial differential equations on manifolds” in general. The theoretical approach presented heavily relies on a the framework on differential forms and “modern” differential geometry (Burke, 1996; Frankel, 2011; Bossavit, 2012). They are well known in certain communities, e.g. modern finite element theory (Arnold et al., 2006), but not so much in continuum mechanics where classical vector and tensor calculus are predominant. Notable exceptions are the works of Hirani (2003), Leok (2004) and Kanso et al. (2007). Hence, a considerable amount of work has gone into the introduction of this framework which is much richer in geometric objects than classical tensor and vector calculus. As a reward one gains a well-defined set of rules governing the operations on the objects encountered and clear geometrical interpretations that even give hints on the proper finite element discretization. Thus, a main task in the “new” framework is to find the appropriate mathematical object for the representation of a physical quantity. This automatically fixes the rules. By contrast, in tensor calculus things are not always that clear and it is thus much easier to unconsciously leave the path of physically reasonable mathematical operations. The thesis however does not stop at the translation of basic magnetic, electric and mechanical quantities to a new framework but goes further: invariants of tensorial quantities and forms in connection with the principles of material frame indifference and covariance are one of main topics waiting for further extension in future publications. Connected to that, the thesis contains a discussion of different parameterizations of energy functions, their time rates and relations between them.
Appears in Collections:02 Fakultät Bau- und Umweltingenieurwissenschaften

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