Forschungsprojekte

We present the derivation of a simple viscous damping model of Kelvin-Voigt type for geometrically exact Cosserat rods from three-dimensional continuum theory. Assuming moderate curvature of the rod in its reference configuration, strains remaining small in its deformed configurations, strain rates that vary slowly compared to internal relaxation processes, and a homogeneous and isotropic material, we obtain explicit formulas for the damping parameters of the model in terms of the well known stiffness parameters of the rod and the retardation time constants defined as the ratios of bulk and shear viscosities to the respective elastic moduli. We briefly discuss the range of validity of the Kelvin-Voigt model and illustrate its behaviour for large bending deformations with a numerical example.

 

 

The theory of Cosserat rods provides versatile models for the simulation of large spatial deformations of slender flexible structures. As the strain measures of the mechanical theory are given in terms of the differential invariants of Cosserat curves, the kinematics of Cosserat rods are closely related to the differential geometry of framed curves. We utilize ideas from the difference geometry of framed curves in Euclidean space to construct the discrete kinematics of Cosserat rod models on a staggered grid, preserving the essential geometric properties independent of the coarseness of the discretization. On this basis, we also derive a vertex based model variant and discuss its relation to the discrete Cosserat rod kinematics obtained by a geodesic interpolation of the nodal variables in the Lie groups E 3 × SO(3) and SE(3).

The theory of Cosserat rods provides a self consistent framework for modeling large spatial deformations of slender flexible structures at small local strains. Discrete Cosserat rod models [1, 2] based on geometric finite differences preserve essential properties of the continuum theory. The present work investigates kinetic aspects of discrete quaternionic Cosserat rods defined on a staggered grid within the framework of Lagrangian mechanics. Assuming hyperelastic constitutive behaviour, the Euler–Lagrange equations of the model are shown to be equivalent to the (semi)discrete balance equations of forces, moments and inertial terms obtained from a direct discretization of the continuous balance equations via spatial finite differences along the centerline curve. Therefore, equilibrium configurations obtained by energy minimization correspond to solutions of the quasi-static balance equations. We illustrate this approach by two academic examples (Euler’s Elastica and Kirchhoff’s helix) and highlight its utility for practical applications with a use case from automotive industry (analysis of the layout of cooling hoses in the engine compartment of a passenger car).

For software tools currently used in industry for computer aided design (CAD), digital mock-up and virtual assembly there is an increasing demand to handle not only rigid geometries, but to provide also capabilities for realistic simulations of large deformations of slender flexible structures in real time (i.e.: at interactive rates). The theory of Cosserat rods provides a framework to perform physically correct simulations of arbitrarily large spatial deformations of such structures by stretching, bending and twisting. The kinematics of Cosserat rods is described by the differential geometry of framed curves, with the differential invariants of rod configurations corresponding to the strain measures of the mechanical theory. We utilize ideas from the discrete differential geometry of framed curves in combination with the variational framework of Lagrangian mechanics to construct discrete Cosserat rod models that behave qualitatively correct for rather coarse discretizations, provide a fast computational performance at moderate accuracy, and thus are suitable for interactive simulations. This geometry based discretization approach for flexible 1D structures has industrial applications in design and digital validation. We illustrate this with some application examples from automotive industry.

The theory of Cosserat rods provides versatile models to simulate large spatial deformations of slender flexible structures. As the strain measures of the mechanical theory are given in terms of the differential invariants of Cosserat curves, the kinematics of Cosserat rods is closely related to the differential geometry of framed curves. We utilize ideas from the difference geometry of framed curves in Euclidean space to construct the discrete kinematics of Cosserat rod models on a staggered grid, preserving the essential geometric properties independent of the coarseness of the discretization.