A New Kirchhoff-Love Beam Element and its Application to Polymer Mechanics

The novel finite element formulations fall into the category of  geometrically exact Kirchhoff-Love beams. A prominent characteristic of  this category is that the absence of shear deformation is strongly  enforced by removing two degrees of freedom. Further, the corresponding  beam theories exhibit not only translational but also rotational degrees  of freedom and their configurations thus form a non-additive and  non-commutative space. Sophisticated interpolation schemes are required  that need to be tested not only for locking, spatial convergence  behavior, and energy conservation, but also for observer invariance and  path-independence. For the three novel beam element formulations all  these properties are analytically and numerically studied and confirmed,  if applicable. Two different rotation parameterization strategies are  employed based on the well-known geodesic interpolation used in many  Simo-Reissner beams and the lesser known split into the so-called  \textit{smallest rotation} and a torsional part. Application of the  former parameterization results in a mixed finite element formulation  intrinsically free of locking phenomena. Additionally, the first  geometrically exact Kirchhoff-Love beam element is presented, which  strongly enforces inextensibility by removing another degree of freedom.  Furthermore, the numerical efficiency of the new beam formulations is  compared to other beam elements that allow for or suppress shear  deformation. When modeling very slender beams, the new elements offer  distinct numerical advantages.

Standard molecular dynamics  simulations, which are commonly used to study polymers, suffer from a  lack of a careful mathematical basis and the use of an expensive  explicit time integration scheme. To circumvent these shortcomings and  to be able to simulate stretching experiments on relevant time scales,  the problem is described by a stochastic partial differential equation,  which can be solved using the finite element method with a backward  Euler temporal discretization. In detail, the polymer is represented by a  Kirchhoff-Love beam with a linear elastic constitutive model. Inertial  and electrostatic forces are neglected. It is deformed by a distributed  load mimicking collisions with molecules of the surrounding fluid.  Naturally, this load heavily fluctuates over time and space and mean  values need to be computed in a Monte Carlo manner. To vastly speed up  the fitting process to experimental data in a Bayesian framework, a  surrogate model based on a Gaussian process is set up, which directly  computes the mean values for given material parameters. The  uncertainties and correlations of the material parameters are studied  and compared to the literature.