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, \(\rho = 100.0\) \(\mathrm{kg/m^3}\).
   • (iii) Cross-sectional radius, \(r_0 = 0.01\mathrm{~m}\).
   • (iv) Damping viscosity, \(\mu = 1.0\).
   • (v) Gravity, \(\mathbf{g} = [0.0, 0.0]^T\) \(\mathrm{m/s^2}\).
   • (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 \(\dot{F} = 10\mathrm{~N/s}\).

Dynamic Rendering