Keywords: cable, element, hyperelasticity, minimization, nonlinear, optimization..

This paper presents a mathematical model of a line element for hyperelastic finite element analysis. The deformation gradient tensor is written in terms of nodal displacements. The invariants of the right Cauchy-Green deformation tensor are written in terms of nodal displacements. The total potential energy is minimized using a quasi-Newton method.

In the case of an incompressible material, the incompressibility constraint is satisfied exactly avoiding difficulties in the numerical simulation. The boundary condition implies that the stress orthogonal to the element's line is equal to zero and the thickness of the element in the deformed state can be removed from the expressions of the invariants of the right Cauchy-Green deformation tensor. In the case of a compressible material, it is shown that the minimum of the total potential energy with respect to the element thickness implies that the principal stress orthogonal to the element's line is zero.

The use of a quasi-Newton method to minimize the total potential energy function has several advantages over solving the equilibrium equations in nonlinear mechanics. It allows the analysis of under constrained structures even without support constraints to prevent rigid body motion. It is not necessary to derive the tangent stiffness matrix. It is not necessary to solve any system of equations. It can handle large scale problems with efficiency.

The computer code uses the limited memory BFGS to handle large scale problems and a line search procedure with safeguards.

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