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Geometric electroelasticity
(2014)
In this work a diffential geometric formulation of the theory of electroelasticity is developed which also includes thermal and magnetic influences. We study the motion of bodies consisting of an elastic material that are deformed by the influence of mechanical forces, heat and an external electromagnetic field. To this end physical balance laws (conservation of mass, balance of momentum, angular momentum and energy) are established. These provide an equation that describes the motion of the body during the deformation. Here the body and the surrounding space are modeled as Riemannian manifolds, and we allow that the body has a lower dimension than the surrounding space. In this way one is not (as usual) restricted to the description of the deformation of three-dimensional bodies in a three-dimensional space, but one can also describe the deformation of membranes and the deformation in a curved space. Moreover, we formulate so-called constitutive relations that encode the properties of the used material. Balance of energy as a scalar law can easily be formulated on a Riemannian manifold. The remaining balance laws are then obtained by demanding that balance of energy is invariant under the action of arbitrary diffeomorphisms on the surrounding space. This generalizes a result by Marsden and Hughes that pertains to bodies that have the same dimension as the surrounding space and does not allow the presence of electromagnetic fields. Usually, in works on electroelasticity the entropy inequality is used to decide which otherwise allowed deformations are physically admissible and which are not. It is alsoemployed to derive restrictions to the possible forms of constitutive relations describing the material. Unfortunately, the opinions on the physically correct statement of the entropy inequality diverge when electromagnetic fields are present. Moreover, it is unclear how to formulate the entropy inequality in the case of a membrane that is subjected to an electromagnetic field. Thus, we show that one can replace the use of the entropy inequality by the demand that for a given process balance of energy is invariant under the action of arbitrary diffeomorphisms on the surrounding space and under linear rescalings of the temperature. On the one hand, this demand also yields the desired restrictions to the form of the constitutive relations. On the other hand, it needs much weaker assumptions than the arguments in physics literature that are employing the entropy inequality. Again, our result generalizes a theorem of Marsden and Hughes. This time, our result is, like theirs, only valid for bodies that have the same dimension as the surrounding space.
We study buckling instabilities of filaments in biological systems. Filaments in a cell are the building blocks of the cytoskeleton. They are responsible for the mechanical stability of cells and play an important role in intracellular transport by molecular motors, which transport cargo such as organelles along cytoskeletal filaments. Filaments of the cytoskeleton are semiflexible polymers, i.e., their bending energy is comparable to the thermal energy such that they can be viewed as elastic rods on the nanometer scale, which exhibit pronounced thermal fluctuations. Like macroscopic elastic rods, filaments can undergo a mechanical buckling instability under a compressive load. In the first part of the thesis, we study how this buckling instability is affected by the pronounced thermal fluctuations of the filaments. In cells, compressive loads on filaments can be generated by molecular motors. This happens, for example, during cell division in the mitotic spindle. In the second part of the thesis, we investigate how the stochastic nature of such motor-generated forces influences the buckling behavior of filaments. In chapter 2 we review briefly the buckling instability problem of rods on the macroscopic scale and introduce an analytical model for buckling of filaments or elastic rods in two spatial dimensions in the presence of thermal fluctuations. We present an analytical treatment of the buckling instability in the presence of thermal fluctuations based on a renormalization-like procedure in terms of the non-linear sigma model where we integrate out short-wavelength fluctuations in order to obtain an effective theory for the mode of the longest wavelength governing the buckling instability. We calculate the resulting shift of the critical force by fluctuation effects and find that, in two spatial dimensions, thermal fluctuations increase this force. Furthermore, in the buckled state, thermal fluctuations lead to an increase in the mean projected length of the filament in the force direction. As a function of the contour length, the mean projected length exhibits a cusp at the buckling instability, which becomes rounded by thermal fluctuations. Our main result is the observation that a buckled filament is stretched by thermal fluctuations, i.e., its mean projected length in the direction of the applied force increases by thermal fluctuations. Our analytical results are confirmed by Monte Carlo simulations for buckling of semiflexible filaments in two spatial dimensions. We also perform Monte Carlo simulations in higher spatial dimensions and show that the increase in projected length by thermal fluctuations is less pronounced than in two dimensions and strongly depends on the choice of the boundary conditions. In the second part of this work, we present a model for buckling of semiflexible filaments under the action of molecular motors. We investigate a system in which a group of motors moves along a clamped filament carrying a second filament as a cargo. The cargo-filament is pushed against the wall and eventually buckles. The force-generating motors can stochastically unbind and rebind to the filament during the buckling process. We formulate a stochastic model of this system and calculate the mean first passage time for the unbinding of all linking motors which corresponds to the transition back to the unbuckled state of the cargo filament in a mean-field model. Our results show that for sufficiently short microtubules the movement of kinesin-I-motors is affected by the load force generated by the cargo filament. Our predictions could be tested in future experiments.