Jonas Harsch

Zusammenfassung

Based on more than three decades of rod finite element theory, this publication combines the successful contributions found in the literature and eradicates the arising drawbacks like loss of objectivity, locking, path-dependence and redundant coordinates. Specifically, the idea of interpolating the nodal orientations using relative rotation vectors, proposed by Crisfield and Jelenic in 1999, is extended to the interpolation of nodal Euclidean transformation matrices with the aid of relative twists; a strategy that arises from the SE(3)-structure of the Cosserat rod kinematics. Applying a Petrov--Galerkin projection method, we propose a rod finite element formulation where the virtual displacements and rotations as well as the translational and angular velocities are interpolated instead of using the consistent variations and time-derivatives of the introduced interpolation formula. Properties such as the intrinsic absence of locking, preservation of objectivity after discretization and parameterization in terms of a minimal number of nodal unknowns are demonstrated by representative numerical examples in both statics and dynamics.

Zusammenfassung

The standard in rod finite element formulations is the Bubnov–Galerkin projection method, where the test functions arise from a consistent variation of the ansatz functions. This approach becomes increasingly complex when highly nonlinear ansatz functions are chosen to approximate the rod's centerline and cross-section orientations. Using a Petrov–Galerkin projection method, we propose a whole family of rod finite element formulations where the nodal generalized virtual displacements and generalized velocities are interpolated instead of using the consistent variations and time derivatives of the ansatz functions. This approach leads to a significant simplification of the expressions in the discrete virtual work functionals. In addition, independent strategies can be chosen for interpolating the nodal centerline points and cross-section orientations. We discuss three objective interpolation strategies and give an in-depth analysis concerning locking and convergence behavior for the whole family of rod finite element formulations.

Zusammenfassung

Making use of experimental data for bias extension, shearing, and point-load tests in large deformation regime for rectangular and square bi-pantographic specimens, we perform a numerical identification to fit the a priori parameters of a planar discrete spring model. The main objective of the work is to develop an automatized optimization process based on the Nelder�Mead simplex algorithm for identifying the constitutive parameters of discrete modeling of bi-pantographic structures, as well as assessing its descriptiveness and predictive capacity. The analysis allows to conclude that there exists a single set of parameters for the adopted discrete modeling such that it is descriptive and predictive for several different tests and for a wide range of deformations.

Zusammenfassung

To describe the mechanical behavior of a soft pneumatic actuator, we present a nonlinear rod theory, which recognizes the chamber pressurization as contribution to the resultant contact forces and couples. Accordingly, the pressure actuation can be considered as part of the rod’s constitutive laws, which relate the contact interactions with the rod’s kinematics and the chamber pressures. The theory allows for nonlinear constitutive laws, which are capable of describing strengthening or softening behavior of largely deformable materials. The theory was applied to a three-chamber actuator to predict its centerline as well as the cross-section orientations in static equilibrium. The resulting governing equations were spatially discretized using beam finite elements. We evaluated four different actuator models that are contained in the presented theory regarding describability and predictability of the end effector position. Existing rod models with linear elastic material laws can describe the tip-displacement with an averaged deviation of 9.1 mm in the training set and 11.4 mm in the test set. With a deviation of 2.8 mm and 4.9 mm in the training and test set, respectively, the most enhanced model, for which the pressure chamber radii increase with increasing pressure, is more than 2 times more accurate.

Zusammenfassung

More than 100 years ago, Lionel Robert Wilberforce did investigations On the Vibrations of a Loaded Spiral Spring 1. The spring was clamped at its upper side and on the other side, perpendicular to the spring axis, a steel cylinder was attached. Four screws with adjustable nuts were symmetrically attached around the cylinder in order to change its moment of inertia (Fig. 1). In this paper the Wilberforce pendulum is modeled by a rigid body attached to a constrained spatial nonlinear Timoshenko beam, discretized with B-spline shape functions. As shown by a numerical experiment, the presented model is capable of reproducing the characteristic pendulum motion.

Zusammenfassung

A new director-based finite element formulation for geometrically exact beams is proposed by weak enforcement of the orthonormality constraints of the directors. In addition to an improved numerical performance, this formulation enables the development of two more beam theories by adding further constraints. Thus, the paper presents a complete intrinsic spatial nonlinear theory of three kinematically different beams which can undergo large displacements and which can have precurved reference configurations. Moreover, the hyperelastic constitutive laws allow for elastic finite strain material behavior of the beams. Furthermore, the numerical discretization using concepts of isogeometric analysis is highlighted in all clarity. Finally, all presented models are numerically validated using exclusive analytical solutions, existing finite element formulations, and a complex dynamical real-world example.

Zusammenfassung

This article is based on the planar beam theories presented in Eugster and Harsch (2020) and deals with the finite element analysis of their presented beam models. A Bubnov-Galerkin method, where B-splines are chosen for both ansatz and test functions, is applied for discretizing the variational formulation of the beam theories. Five different planar beam finite element formulations are presented: The Timoshenko beam, the Euler--Bernoulli beam obtained by enforcing the cross-section's orthogonality constraint as well as the inextensible Euler--Bernoulli beam by additionally blocking the beam's extension. Furthermore, the Euler--Bernoulli beam is formulated with a minimal set of kinematical descriptors together with a constrained version that satisfies inextensibility. Whenever possible, the numerical results of the different formulations are compared with analytical and semi-analytical solutions. Additionally, numerical results reported in classical beam finite element literature are collected and reproduced.

Zusammenfassung

This article intends to present a concise theory of spatial nonlinear classical beams followed by a special treatment of the planar case. Hereby the considered classical beams are understood as generalized one-dimensional continua that model the mechanical behavior of three-dimensional beam-like objects. While a onedimensional continuum corresponds to a deformable curve in space, parametrized by a single material coordinate and time, a generalized continuum is augmented by further kinematical quantities depending on the very same parameters. We introduce the following three nonlinear spatial beams: The Timoshenko beam, the Euler--Bernoulli beam and the inextensible Euler--Bernoulli beam. In the spatial theory, the Euler--Bernoulli beam and its inextensible companion are presented as constrained theories. In the planar case, both constrained theories are additionally described using an alternative kinematics that intrinsically satisfies the defining constraints of these theories.

Zusammenfassung

A 2D-continuum model describing finite deformations in plane of discrete bi-pantographic fabrics has been recently obtained by applying an asymptotic procedure based on a set of local generalized coordinates. Rectangular bi-pantographic prototypes were additively manufactured by selective laser sintering using polyamide as raw material. Displacement-controlled bias extension tests were performed on such specimens for total elastic deformations up to ca. 25\%. Experimental force measurements, complemented by discrete displacement measurements obtained by local digital image correlation, were used to fit the continuum model. In the present paper, a global and minimal set of generalized coordinates, alternative to the one used for the homogenization, is introduced for the discrete model. The mechanical constitutive parameters appearing in the discrete model are then found by means of collected experimental data. Finally, a comparison between experiments, the discrete and the continuum model is presented. It is concluded that (a) the discrete model and the experimental data are in excellent agreement, and that (b) the continuum retains the relevant phenomenology of the discrete system even for a rather low number of cells.