### Filtern

#### Schlagworte

In this article, we consider the quasistatic boundary value problems of linear elasticity and nonlinear elastoplasticity, with linear Hooke’s law in the elastic regime for both problems and with the linear kinematic hardening law for the plastic regime in the latter problem. We derive expressions and estimates for the difference of the solutions of both models, i.e. for the stresses, the strains and the displacements. To this end, we use the stop and play operators of nonlinear functional analysis. Further, we give an explicit example of a homotopy between the solutions of both problems.

In this paper, we present a viscoelastic rod model that is suitable for fast and sufficiently accurate dynamic simulations. It is based on Cosserat’s geometrically exact theory of rods and is able to represent extension, shearing (’stiff ’ dof), bending and torsion (’soft’ dof). For inner dissipation, a consistent damping potential from Antman is chosen. Our discrete model is based on a finite difference discretisation on a staggered grid. The right-hand side function f and the Jacobian ∂f/∂(q, v, t) of the dynamical system q˙ = v, v˙ = f(q, v, t) – after index reduction from three to zero – is free of higher algebraic (e.g. root) or transcendent (e.g. trigonometric or exponential) functions and is therefore cheap to evaluate. For the time integration of the system, we use well established stiff solvers like RADAU5 or DASPK. As our model yields computation times within milliseconds, it is suitable for interactivemanipulation in ’virtual reality’ applications. In contrast to fast common VR rod models, our model reflects the structural mechanics solutions sufficiently correct, as comparison with ABAQUS finite element results shows.

In this paper, the model of Köttgen, Barkey and Socie, which corrects the elastic stress and strain tensor histories at notches of a metallic specimen under non-proportional loading, is improved. It can be used in connection with any multiaxial s -e -law of incremental plasticity. For the correction model, we introduce a constraint for the strain components that goes back to the work of Hoffmann and Seeger. Parameter identification for the improved model is performed by Automatic Differentiation and an established least squares algorithm. The results agree accurately both with transient FE computations and notch strain measurements.

Classical geometrically exact Kirchhoff and Cosserat models are used to study the nonlinear deformation of rods. Extension, bending and torsion of the rod may be represented by the Kirchhoff model. The Cosserat model additionally takes into account shearing effects. Second order finite differences on a staggered grid define discrete viscoelastic versions of these classical models. Since the rotations are parametrised by unit quaternions, the space discretisation results in differential-algebraic equations that are solved numerically by standard techniques like index reduction and projection methods. Using absolute coordinates, the mass and constraint matrices are sparse and this sparsity may be exploited to speed-up time integration. Further improvements are possible in the Cosserat model, because the constraints are just the normalisation conditions for unit quaternions such that the null space of the constraint matrix can be given analytically. The results of the theoretical investigations are illustrated by numerical tests.

In this article, we summarise the rotation-free and quaternionic parametrisation of a rigid body. We derive and explain the close interrelations between both parametrisations. The internal constraints due to the redundancies in the parametrisations, which lead to DAEs, are handled with the null space technique. We treat both single rigid bodies and general multibody systems with joints, which lead to external joint constraints. Several numerical examples compare both formalisms to the index reduced versions of the corresponding standard formulations.

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 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 our damping model and illustrate
its behaviour with a numerical example.