Hyperelastic planar cable

In this subsection, we examine a simple planar cable, where, unlike a bending-dominated beam primarily governed by bending energy, the configuration is mainly determined by stretching energy. Mathematically, for a 1D structure of length $L$, with linear elastic stretching stiffness $EA$, bending stiffness $EI$, and subjected to an external load $F$, a beam model is appropriate when $EA\gg F\sim EI/L^2$, whereas a cable model should be used when $EA\sim F\gg EI/L^2$. In an intermediate scenario, where $EA\gg F\gg EI/L^2$, the configuration of the structure is governed solely by its geometric characteristics and boundary conditions, rendering the problem material-independent — such as an inextensible catenary under its self-weight, which is known as catenary. On the other hand, for sufficiently large external loads, the material response may exceed the linear regime, necessitating the use of a hyperelastic model.

Simulation Initialization

To initialize the simulation, the following inputs are used:

1. Geometry and connection:
   • (i) Nodal positions: the position of the nodes $\mathbf{q}(t=0)$, with a total of $N=40$. The length of the cable is set at $L=1.0$ m.
   • (ii) Stretching elements: the connections between the nodes, with a total of $N_s=39$.
2. Physical parameters:
   • (i) Young’s modulus, $E=1.0\mathrm{MPa}$, $C_1=4E/30$, and $C_2=E/30$.
   • (ii) Material density, $\rho =100.0$ $\mathrm{kg}/{\mathrm{m}}^3$.
   • (iii) Cross-sectional radius, $r_0=0.01\mathrm{m}$.
   • (iv) Damping viscosity, $\mu =1.0$.
   • (v) Gravity, $\mathbf{g}=[0.0,0.0{]}^T$ $\mathrm{m}/{\mathrm{s}}^2$.
   • (vi) The overall simulation is dynamic, i.e., $\mathrm{ifStatic}=0$.
3. Numerical parameters:
   • (i) Total simulation time, $T=1.0\mathrm{s}$.
   • (ii) Time step size, $\mathrm{dt}=0.001\mathrm{s}$.
   • (iii) Numerical tolerance, $\mathrm{tol}=1\times {10}^{-4}$.
   • (iv) Maximum iterations, $N_{\mathrm{iter}}=10$.
4. Boundary conditions:
   • The first node and the last node, $\{{\mathbf{x}}_1,{\mathbf{x}}_{40}\}$, are fixed to achieve a pin-pin boundary condition, thus the constrained array, $\mathcal{FIX}=[1,2,79,80{]}^T$.
5. Initial conditions:
   • (i) Initial position is input from the nodal positions.
   • (ii) Initial velocity is set to zeros.
6. Loading steps:
   • The external vertical force is increased at a rate $\dot{F}=10\mathrm{N}/\mathrm{s}$.

Dynamic Rendering