PAMM · Proc. Appl. Math. Mech. 17, 293 – 294 (2017) / DOI 10.1002/pamm.201710116
Bending of Viscoplastic Cables
, Joachim Linn
, and Stefan Diebels
Fraunhofer Institute for Industrial Mathematics, Dept. Mathematical Methods in Dynamics and Durablitiy, Fraunhofer
Platz 1, D-67663 Kaiserslautern
Universität des Saarlandes, Lehrstuhl für Technische Mechanik, Campus A4.2, D-66123 Saarbrücken
This contribution deals with the experimental investigation of cables showing inelastic behavior under bending deformation.
A new experimental device, which enables the direct measurement of the quantities entering the constitutive law for bending,
i.e. the bending moment and bending curvature, is introduced. Results from advanced cyclic experiments executed on this
new pure bending device show inelastic effects. It is remarkable, that this procedure allows for the distinction of intrinsic
inelastic effects, e.g. due to friction between the constituents, from global inelastic effects such as damage of the specimen.
2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction & Theory
Cables are ﬂexible components with a high aspect ratio. Their deformation behavior can thus be modeled within the framework
of Cosserat rod theory , which uses geometrically exact kinematics relating conﬁguration variables and objective strain
measures, balance equations in terms of the sectional quantities and constitutive equations. The present work focuses on the
latter, since constitutive laws yield the possibility to adapt the model to various kinds of observed deformation behavior. In
case of the Cosserat rod model, the constitutive equations can be formulated in the material sectional forces F and the material
sectional moments M. In the linear elastic case, they are related to the corresponding objective deformation measures via
F = C
· Γ; M = C
· K, (1)
where Γ denotes the vector of the translational deformation measures (shear strains and axial strain), K the vector of rotational
deformation measures (bending curvatures and torsional twist) and C
the effective stiffness matrices.
Most cables’ cross sections can be assumed to be circular. However, they are not homogeneous, because cables consist
at least of two different components such as metallic wires, rubber jackets or other reinforcements. Therefore, inelastic
effects caused by the structural setup, e.g. delamination, friction or damage, have to be expected when cables are deformed.
Additionally, inelastic material effects, resulting from inelastic behavior of the constituents, inﬂuence the deformation behavior
of cables. These effects have been shown for simple cables in [2, 3] using classical uniaxial tensile tests, torsion tests and
three point bending tests executed in a cyclic procedure. The present work focuses on inelastic bending behavior of cables
since this is the load case most relevant in large deformation applications of cables. The linear elastic constitutive law for
planar bending can be formulated in the effective bending curvature K
and the effective bending moment M
, using the
linear elastic bending stiffness (EI)
, i.e. M
. Advanced bending experiments on cables showing inelastic
behavior are executed in order to investigate their constitutive behavior in more detail.
2 Bending Experiments for Cables
The three point bending experiment is state of the art for the determination of the linear elastic bending stiffness of beam-like
specimens such as cables. It is well-known and easily implemented, but it has some drawbacks in regard of the investigation
of ﬁnite deformations. Most importantly, the analytical solution used to derive the bending stiffness is only valid for small
deﬂections of the specimen. Furthermore, the real bearing situation is different from the theoretical boundary conditions. Fric-
tion in the supports causes normal and shear forces on the specimen, which distort the experimental results. The experimental
setup of three point bending yields a deformation state where the bending moment increases linearly from the ends of the
specimen to the middle and the curvature is not constant along the cable axis. The quantities entering the constitutive law for
bending, i.e. bending curvature and bending moment, are therefore not directly accessible in three point bending.
A new experimental setup for bending of cables, which enables direct access to the bending moment and bending curvature,
was designed. A deformation state of pure bending is achieved by applying only a bending moment on the specimen. The ex-
perimental setup is explained in ﬁgure 1. It ensures that no normal or shear forces act on the specimen during the experiment.
The centerline of the specimen is bend into a circular arc with constant bending curvature, and consequently constant bending
moment, along the specimen.
In order to observe inelastic effects, the pure bending experiment is performed in consecutive cycles, as described for the
classical experiments in . In the following, results of experiments executed on a simple cable, consisting of inner aluminum
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2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim