Towards realistic nonlinear elastic bending behavior for cable simulation

This contribution aims to characterize and model the nonlinear elastic behavior of cables for reliable simulations. To simulate the nonlinear elastic behavior of cables, we use an iterative method, where at each step, an algorithmic local bending stiffness constant is used and updated according to the current cable state. We also formulate an inverse problem to determine the properties of real cables. By solving the inverse problem, the nonlinear elastic behavior for given measurement data is identified, yielding a curvature‐dependent bending stiffness characteristic. In addition, we propose an alternative method based on the balance equations for rods in static equilibrium to identify the bending stiffness characteristic. We apply both methods to experimental data, and the results are compared and discussed.


INTRODUCTION
Flexible structures, such as cables and hoses, are widely used in the automotive industry [1].The demand to reliably simulate such structures is, therefore, increased.Cosserat rod theory provides a suitable framework to model such flexible structures in a geometrically exact way.Linn et al. [2,3] described a robust and efficient framework, in which the static equilibrium of the Cosserat rod is calculated by minimizing the potential energy.However, in these works, only linear elasticity was considered.For cables with more complex structures and materials, nonlinear elasticity plays an important role.We can observe such nonlinear elasticity in both pure bending tests [4] and MeSOMICS bending test [5].In this work, we use a curvature-dependent bending stiffness characteristic to represent the nonlinear elastic bending behavior.To simulate such behavior, we presented an iterative method [6] based on the framework in Linn et al. [2].In each iteration, the static equilibrium is calculated by minimizing the potential energy for linear elastic behavior, where an algorithmic local bending stiffness constant is included.The algorithmic local bending stiffness constant is updated according to a given curvature-dependent bending stiffness characteristic, and the iteration is repeated until the convergence of the cable state is achieved.Furthermore, we present two methods to determine the bending stiffness characteristics of real cables: an inverse problem and a method based on the equilibrium equations of the Cosserat rod.Since only quasi-static experiments were considered to date, dynamic effects such as damping are not considered in the simulations presented in this work.

COSSERAT ROD IN TWO-DIMENSIONAL SPACE
In this section, we show a brief introduction of the Cosserat rod in two-dimensional space, which was introduced in our previous works in Zhao et al. [6,7].The Cosserat rod in two-dimensional space (see Figure 1) consists of a centerline ((), ())  ∈ ℝ 2 and the rotation angle () parametrizing the local frame with rotation matrix where the curve parameter  ∈ [0, ] is the arc length of the centerline in the reference configuration of the rod.The curvature of the moving frame, which approximately corresponds to the curvature of the centerline, and the material tangent vector, which contains the components of the centerline tangent vector w.r.t the local frame, are given by The difference functions Δ() = () − K0 () and ΔΓ Γ Γ() = Γ Γ Γ() − Γ Γ Γ 0 () represent the deviation of () and Γ Γ Γ() from the reference values K0 () and Γ Γ Γ 0 = (1, 0)  .Following the work in Linn et al. [2], the static equilibrium state of the cables can be obtained by minimizing the potential energy where the first term represents the shear and tension energy, and the second term represents the bending energy.The effective tension stiffness [] and effective shear stiffness [] are assembled in the diagonal coefficient matrix  Γ Γ Γ = diag([], []) and [] is the effective bending stiffness.The discrete potential energy can be written as where the index  denotes vertex quantities at   for  = 0, … ,  and  + 1∕2 denotes edge-centered quantities at  +1∕2 for  = 0, … ,  − 1.More details about the derivation from the continuous case can be found in Linn et al. [2].

BENDING EXPERIMENTS
Real and virtual bending experiments are performed to study the bending behavior of cables.
The real experiments are performed on the MeSOMICS measurement machine [5].As shown in Figure 2, the specimen is clamped at both ends.Both clamping devices are inserted in (approximately) moment-free bearings.The left clamping point is stepwise shifted towards the right clamping point, resulting in corresponding bending deformations.By moving back to the start point, one bending cycle is achieved.Such cycle repeats five times, and here, we evaluate the fourth cycle.Meanwhile, the resulting reaction force is measured at the right clamping point.For each configuration, the bending deformations are recorded by a camera and the corresponding cable centerline is detected.Figure 2 shows the captured bending deformation of a cable specimen (cable 1) with length  = 181 mm and diameter  = 4.6 mm (cross section as shown in the left picture of Figure 3).The three images in Figure 2 show cable configurations (a), (b), and (c) with applied displacements of , 2, and 3, respectively.Furthermore, we perform the bending experiment on a high-voltage cable specimen (cable 2), which has a length of 226 mm and a diameter of 17.4 mm, the cross section is as shown in the right picture of Figure 3.
We also perform virtual bending experiments in a two-dimensional space by using the Cosserat rod model from Section 2. The boundary conditions of the virtual experiment are the same as those for the real experiment.In the virtual experiment, both ends are moment-free.By applying displacements to one end, the resultant forces on the other end are computed.

SIMULATING NONLINEAR ELASTIC BEHAVIOR
To represent the nonlinear elastic bending behavior, we introduce a curvature-dependent bending stiffness characteristic   ().With that, the bending energy term is written as F I G U R E 3 Left: Cross section of cable 1. Right: Cross section of cable 2.
TA B L E 1 Model parameters for virtual bending experiment.
Using  , as bending energy term, the energy minimization problem could be solved directly.However, this requires expensive computations.To use the curvature-dependent bending stiffness characteristic with Cosserat rod in twodimensional space (as described in Section 2) while maintaining its efficiency, we use an iterative method.In each step, we still consider linear elasticity and solve the static equilibrium by energy minimization.More precisely, an algorithmic local bending stiffness constant is applied here.In the th iteration step, a given cable state ( | <  is fulfilled.More technical details regarding the iterative method can be found in Zhao et al. [6]. We perform virtual bending experiments to demonstrate the iterative method.Here, the curvature-dependent bending stiffness characteristic is parametrized using a natural cubic spline with three control points, denoted by ( κ0 , [] 0 ), ( κ1 , [] 1 ) and ( κ2 , [] 2 ).We also perform a virtual bending experiment with a direct solution for the bending energy term in Equation ( 5) as reference solution to validate the iterative method.The model parameters are listed in Table 1.
The plots in Figure 4 show the discrete curvature   over the arc length   for the cable states in the single iteration steps, while the boundary conditions are unchanged.One can observe that after six iterations, the curvature in the iterative method converges to the curvature values of the reference solution (illustrated by the black dashed line).Figure 5 plots the horizontal reaction force virtually measured at the fixed end point for a sequence of applied displacements.A very good agreement is observed between the horizontal reaction force simulated by iterative method and the reference solution.
Moreover, the iterative method is computationally more efficient (in this example, approximately five times faster) than directly solving the energy minimization problem with Equation (5).

IDENTIFICATION OF NONLINEAR ELASTIC BEHAVIOR
Above, we discussed how to utilize a bending stiffness characteristic, that is, nonlinear elastic behavior, in our cable simulation.Now, our focus shifts to determining the bending stiffness characteristic from given measurement results.
F I G U E 4 Discrete curvatures in each iteration.

F I G U R E 5
Horizontal reaction force obtained at the fixed end point, resulting from varying displacements.

Inverse problem
To determine the bending stiffness characteristic, we formulate an inverse problem (see Zhao et al. [6,7]).Our objective is to determine a bending stiffness characteristic that allows the simulated horizontal reaction force and curvature of the cable to closely align with the actual measured force and the cable curvature obtained from the bending tests.
As described in Section 4, the curvature-dependent bending stiffness characteristic is parametrized by a natural cubic spline with three control points.For simplicity, we assume that κ0 , κ1 , and κ2 are fixed, and consider only [] 0 , [] 1 , and [] 2 as optimization variables.Furthermore, we take into account the influence of the precurvature K0 () by including it as an optimization variable.This is necessary, because on one hand, the cable specimen under investigation is not perfectly straight in the absence of applied loading, but instead slightly bent.On the other hand, the precurvature can also approximate the impact of the plastic curvature due to (cyclic) bending.This additional plastic curvature only occurs in regions of strong bending curvature.So in general, the precurvature K0 () can explicitly depend on the arc length  and vary along the cable.However, in this work, we consider the precurvature as constant, denoted as K0 .Using such a precurvature K0 () as an optimization variable can capture the plastic effect to a certain extent, albeit in a rough approximation.The optimization is performed using a Levenberg-Marquardt method in MATLAB [8].
We apply the real bending experiment data to the inverse problem and aim to find the characteristic   () for a real cable.The bending experiment and the cable used in test is detailed in Section 3.
The identified bending stiffness characteristic for cable 1 is shown in Figure 6.The blue dashed line represents the solution of the inverse problem without consideration of the precurvature, that is, with a priori defined precurvature K0 = 0.
F I G U R E 6 Identified bending stiffness characteristic (with set K0 = 0 and K0 as optimization variable) for cable 1.

F I G U R E 7
Measured and simulated force for cable 1 (blue: with dashed blue characteristic and K0 = 0; orange: with solid orange characteristic and identified K0 ).
The solid orange line shows the solution of the inverse problem obtained when the precurvature is considered as an optimization variable (identified precurvature K0 ≈ 3.1 m −1 ).Looking at the corresponding simulated forces in Figure 7 (orange and blue dashed lines), both characteristics lead to good matches with the measured forces (black dashed line).However, comparing the two identified characteristics (see Figure 6), the solution without consideration of the precurvature exhibits negative values for low curvatures, while considering the precurvature as an optimization leads to a more realistic bending stiffness characteristic.This indicates that it is important to include precurvature in the optimization.Furthermore, the solution of the inverse problem with consideration of precurvature (see Figure 6) shows an almost constant bending stiffness, which is plausible due to the simple structure of this cable.
Furthermore, we evaluated the cable 2, which has a more complex structure.The identified bending stiffness characteristic is shown in Figure 8 (identified precurvature ≈ 3.9 m −1 ).It is clear that for this cable, the bending stiffness varies with its curvature.Moreover, as shown in Figure 9, the simulated force using the identified characteristic and the precurvature is in good agreement with the measured value.
Overall, when applied to experimental data, solving the inverse problem yields physically plausible bending stiffness characteristics.
F I G U R E 8 Identified bending stiffness characteristic for cable 2.

F I G U R E 9
Measured and simulated force for cable 2.

Balance equations and (𝜿, 𝒇 𝑬𝑰 (𝜿))-graph
In order to evaluate the bending stiffness characteristic determined as a solution of the inverse problem, we introduced an alternative approach in Zhao et al. [7], which is based on the balance equations for a Cosserat rod.First, we summarize such a observables-based method as described in Zhao et al. [7].
In two-dimensional space, for the rod in the x-y plane (see Figure 1), it holds   () = ((), (), 0)  ,  () = (  (),   (), 0)  , and () = (0, 0, ())  .Thus, Equation ( 9) can be written as where  is a constant scalar.For our bending experiment (see Figure 2), at both simply supported end points, the bending moment fulfills (0) = 0 and () = 0. Since the relative displacement of the boundary conditions is applied along the x-axis and both ends are moment-free, the reaction force in y-direction vanishes, that is,   = 0.Moreover, we choose the origin, such that  = 0, that is, such that (0) = 0. Thus, Equation (10) simplifies to () = ()  , which represents the bending moment at arc length .When [] and [] are relatively large (see Table 1), the Cosserat model can be approximated as an inextensible Kirchhoff model.Therefore, the curvature of the bending line can be further calculated using () = Therefore, given the measured horizontal reaction force and the optically detected bending line, we can determine the stiffness   (()) at curvature ().Since our bending experiment results in a range of curvatures in the rod, we obtain a ((),   (()))-graph for each configuration.
The experimental data of cable 1, which are applied to the inverse problem, are further used to generate (,   ())graphs.
Figure 10 displays the ((),   (()))-graphs for configurations (a), (b), and (c) (see Figure 11), which correspond to the pictures in Figure 2. We observe good consistency among the three identified graphs, indicating the potential to combined them into one single bending stiffness characteristic.However, when comparing the results with those from the inverse problem (the red line in Figure 10), we observe clear deviations.While the order of magnitude is similar, there are clear qualitative discrepancies.Investigating the source of such discrepancies is a topic of our current research.
F I G U R E 1 1 ((),   (()))-graphs determined by method based on balance equations, together with the   () identified via the inverse problem.SUMMARY AND OUTLOOK In this work, we presented two aspects related to investigating the simulation of nonlinear elastic behavior in cables.First, the iterative method demonstrates that we can correctly and efficiently simulate the nonlinear elastic behavior.Additionally, the corresponding inverse problem allows us to access the experimentally observed nonlinear elastic behavior of cables.The identified characteristic including the influence of the precurvature yields physical plausible results.The observables-based method offers the potential to directly identify the bending stiffness characteristic from the optically detected bending line and measured reaction force.Nevertheless, there are discrepancies between the results from the inverse problem and the observables-based ((),   (()))-graphs, the further comparison of these results remains of Our next step is to understand the source of such discrepancies.

F I G U R E 2
Top-view of real bending experiment on cable 1. Left: Cable configuration (a), with applied displacement .Middle: Cable configuration (b), with applied displacement 2.Right: Cable configuration (c), with applied displacement 3.

F I G U R E 1 0
Simulated bending configurations (a), (b), and (c).
This work is part of the "BordNetzSim3D"-project (funding code: 19S21003C), funded by the German Federal Ministry for Economic Affairs and Climate Action.Open access funding enabled and organized by Projekt DEAL.O R C I D Tian Zhao https://orcid.org/0009-0004-3874-2349R E F E R E N C E S